title

Linear Algebra - Full College Course

description

Learn Linear Algebra in this 20-hour college course. Watch the second half here: https://youtu.be/DJ6YwBN7Ya8
This course is taught by Dr. Jim Hefferon, a professor of mathematics at St Michael's College.
📔 The course follows along with Dr. Hefferon's Linear Algebra text book. The book is available for free: http://joshua.smcvt.edu/linearalgebra/book.pdf
📚 Access additional course resources at: https://hefferon.net/linearalgebra/
🔗 Stephen Chew's Learning How to Learn series: https://www.youtube.com/watch?v=htv6eap1-_M&list=PL85708E6EA236E3DB
🔗 3Blue1Brown's Linear Algebra series: https://www.youtube.com/watch?v=fNk_zzaMoSs&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab
⭐️ Course Contents ⭐️
⌨️ (0:00:00) Introduction to Linear Algebra by Hefferon
⌨️ (0:04:35) One.I.1 Solving Linear Systems, Part One
⌨️ (0:26:08) One.I.1 Solving Linear Systems, Part Two
⌨️ (0:40:56) One.I.2 Describing Solution Sets, Part One
⌨️ (0:54:21) One.I.2 Describing Solution Sets, Part Two
⌨️ (1:02:48) One.I.3 General = Particular + Homogeneous
⌨️ (1:18:33) One.II.1 Vectors in Space
⌨️ (1:35:08) One.II.2 Vector Length and Angle Measure
⌨️ (1:51:31) One.III.1 Gauss-Jordan Elimination
⌨️ (2:00:00) One.III.2 The Linear Combination Lemma
⌨️ (2:44:32) Two.I.1 Vector Spaces, Part One
⌨️ (3:08:12) Two.I.1 Vector Spaces, Part Two
⌨️ (3:33:01) Two.I.2 Subspaces, Part One
⌨️ (3:58:16) Two.I.2 Subspaces, Part Two
⌨️ (4:23:43) Two.II.1 Linear Independence, Part One
⌨️ (4:45:11) Two.II.1 Linear Independence, Part Two
⌨️ (5:03:57) Two.III.1 Basis, Part One
⌨️ (5:23:55) Two.III.1 Basis, Part Two
⌨️ (5:42:34) Two.III.2 Dimension
⌨️ (6:03:24) Two.III.3 Vector Spaces and Linear Systems
⌨️ (6:25:09) Three.I.1 Isomorphism, Part One
⌨️ (6:54:08) Three.I.1 Isomorphism, Part Two
⌨️ (7:21:47) Three.I.2 Dimension Characterizes Isomorphism
⌨️ (7:43:43) Three.II.1 Homomorphism, Part One
⌨️ (8:14:52) Three.II.1 Homomorphism, Part Two
⌨️ (8:30:24) Three.II.2 Range Space and Null Space, Part One
⌨️ (9:00:17) Three.II.2 Range Space and Null Space, Part Two.
⌨️ (9:20:57) Three.II Extra Transformations of the Plane
⌨️ (9:52:06) Three.III.1 Representing Linear Maps, Part One.
⌨️ (10:13:18) Three.III.1 Representing Linear Maps, Part Two
⌨️ (10:34:18) Three.III.2 Any Matrix Represents a Linear Map
⌨️ (10:58:32) Three.IV.1 Sums and Scalar Products of Matrices
⌨️ (11:19:14) Three.IV.2 Matrix Multiplication, Part One
⌨️ Three.IV.2 Matrix Multiplication, Part Two (We accidentally left this section out. Watch it here: https://youtu.be/aWubyx5bBn4)
The following sections are in the second video: https://youtu.be/DJ6YwBN7Ya8
⌨️ Three.IV.3 Mechanics of Matrix Multiplication
⌨️ Three.IV.4 Matrix Inverse, Part One
⌨️ Three.IV.4 Matrix Inverse, Part Two
⌨️ Three.V.1 Changing Vector Representations
⌨️ Three.V.2 Changing Map Representations, Part One
⌨️ Three.V.2 Changing Map Representations, Part Two
⌨️ Three.VI Projection
⌨️ Four.I.1 Determinants
⌨️ Four.I.3 Permutation Expansion, Part One
⌨️ Four.I.3 Permutation Expansion, Part Two
⌨️ Four.I.4 Determinants Exist (optional)
⌨️ Four.II.1 Geometry of Determinants
⌨️ Four.III.1 Laplace's formula for the determinant
⌨️ Five.I.1 Complex Vector Spaces
⌨️ Five.II.1 Similarity
⌨️ Five.II.2 Diagonalizability
⌨️ Five.II.3 Eigenvalues and Eigenvectors, Part One
⌨️ Five.II.3 Eigenvalues and Eigenvectors, Part Two
⌨️ Five.II.3 Geometry of Eigenvalues and Eigenvectors
--
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detail

{'title': 'Linear Algebra - Full College Course', 'heatmap': [{'end': 1681.847, 'start': 1258.937, 'weight': 0.717}, {'end': 4626.376, 'start': 3356.383, 'weight': 0.987}], 'summary': "The full college course on linear algebra covers topics like gauss's method, vector spaces, linear maps, determinants, eigenvalues, and eigenvectors, emphasizing self-study tips, importance of exercises, and systematic approach for transforming systems to obtain solutions.", 'chapters': [{'end': 333.713, 'segs': [{'end': 38.13, 'src': 'embed', 'start': 1.92, 'weight': 7, 'content': [{'end': 2.82, 'text': 'Hi, hi, hello.', 'start': 1.92, 'duration': 0.9}, {'end': 4.421, 'text': "Hi, I'm Jim Heffron.", 'start': 3.3, 'duration': 1.121}, {'end': 10.403, 'text': 'This is a series of videos that are based on my undergraduate linear algebra textbook.', 'start': 4.741, 'duration': 5.662}, {'end': 22.667, 'text': 'The idea here is to enable a person to work through the textbook over the course of what would be sort of a typical semester for an undergraduate first linear algebra course.', 'start': 10.843, 'duration': 11.824}, {'end': 29.31, 'text': "So I have a couple things that I wanted to say by way of introduction, so let's get started.", 'start': 23.408, 'duration': 5.902}, {'end': 38.13, 'text': 'So this, the book covers, and these videos will cover, a standard linear algebra course, undergraduate linear algebra course.', 'start': 30.447, 'duration': 7.683}], 'summary': 'Jim heffron presents a series of videos based on his undergraduate linear algebra textbook to help students work through a typical semester course.', 'duration': 36.21, 'max_score': 1.92, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI1920.jpg'}, {'end': 172.348, 'src': 'embed', 'start': 133.215, 'weight': 0, 'content': [{'end': 137.118, 'text': 'then you open up the book and go through the examples and the theorems and the proofs in the book.', 'start': 133.215, 'duration': 3.903}, {'end': 140.528, 'text': "And then you're ready to do the homework.", 'start': 138.601, 'duration': 1.927}, {'end': 143.497, 'text': "In the homework, I've checkmarked some of the exercises.", 'start': 140.969, 'duration': 2.528}, {'end': 145.303, 'text': 'I would do at least those.', 'start': 143.517, 'duration': 1.786}, {'end': 147.293, 'text': 'The answers are available.', 'start': 146.072, 'duration': 1.221}, {'end': 150.215, 'text': 'You can download the answers from the same place you download the book from.', 'start': 147.453, 'duration': 2.762}, {'end': 152.316, 'text': 'Again, look in the description of this video.', 'start': 150.275, 'duration': 2.041}, {'end': 155.137, 'text': "And so you can verify that you've done them right.", 'start': 152.776, 'duration': 2.361}, {'end': 157.639, 'text': 'The answers are completely worked.', 'start': 155.498, 'duration': 2.141}, {'end': 160.281, 'text': 'So if you get stumped on a question, you can.', 'start': 157.759, 'duration': 2.522}, {'end': 163.502, 'text': 'after trying it, you can go and look and see.', 'start': 160.281, 'duration': 3.221}, {'end': 165.584, 'text': 'oh, I see, I should have done okay.', 'start': 163.502, 'duration': 2.082}, {'end': 172.348, 'text': 'so the answers are completely worked, so they should be usable to a person who is on their own, okay?', 'start': 165.584, 'duration': 6.764}], 'summary': 'Use book examples, checkmarked exercises, and downloadable answers for homework verification.', 'duration': 39.133, 'max_score': 133.215, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI133215.jpg'}], 'start': 1.92, 'title': 'Undergraduate linear algebra course overview', 'summary': "Outlines a series of videos based on an undergraduate linear algebra textbook, covering topics like gauss's method, vector spaces, linear maps, determinants, eigenvalues, and eigenvectors, providing a comprehensive overview of the standard course. it also emphasizes effective self-study tips and the importance of doing homework for learning linear algebra.", 'chapters': [{'end': 53.495, 'start': 1.92, 'title': 'Undergraduate linear algebra course overview', 'summary': "Outlines a series of videos based on an undergraduate linear algebra textbook, covering the standard linear algebra course including topics like gauss's method, vector spaces, linear maps, homomorphisms, determinants, eigenvalues, and eigenvectors.", 'duration': 51.575, 'highlights': ['The series of videos are based on an undergraduate linear algebra textbook, aimed to enable a person to work through the textbook over the course of a typical semester for an undergraduate first linear algebra course.', "The textbook and videos cover a standard undergraduate linear algebra course, including topics such as Gauss's method, vector spaces, linear maps, homomorphisms, determinants, eigenvalues, and eigenvectors.", "The course content covers various topics including Gauss's method, vector spaces, linear maps, homomorphisms, determinants, and eigenvalues, which are typical in an undergraduate linear algebra course."]}, {'end': 172.348, 'start': 53.835, 'title': 'Effective self-study tips for undergraduate courses', 'summary': 'Discusses effective self-study tips for undergraduate courses, including the use of videos and textbooks, providing a recommended resource, and the process of watching videos, studying the book, and completing homework.', 'duration': 118.513, 'highlights': ['The process of self-study involves watching videos, studying the book, and completing homework, including checkmarked exercises with available answers.', 'Recommendation of a resource for more effective self-study, with a link provided in the description of the video.', 'Flexibility of using the videos as part of a standard course, as a supplement, or for self-study, with the textbook freely available for self-study purposes.']}, {'end': 333.713, 'start': 173.981, 'title': 'Importance of doing homework', 'summary': 'Emphasizes the importance of doing homework in learning linear algebra, highlighting the necessity of complementing video resources with practical exercises to truly grasp the material.', 'duration': 159.732, 'highlights': ['The importance of doing homework is emphasized as critical in truly learning linear algebra.', 'The chapter cautions against solely relying on watching videos and emphasizes the necessity of completing exercises to learn the material.', "The chapter recommends 3Blue1Brown's linear algebra videos on YouTube, but underscores the need to complement video resources with homework for effective learning."]}], 'duration': 331.793, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI1920.jpg', 'highlights': ["The textbook and videos cover a standard undergraduate linear algebra course, including topics such as Gauss's method, vector spaces, linear maps, homomorphisms, determinants, eigenvalues, and eigenvectors.", 'The series of videos are based on an undergraduate linear algebra textbook, aimed to enable a person to work through the textbook over the course of a typical semester for an undergraduate first linear algebra course.', "The course content covers various topics including Gauss's method, vector spaces, linear maps, homomorphisms, determinants, and eigenvalues, which are typical in an undergraduate linear algebra course.", 'The process of self-study involves watching videos, studying the book, and completing homework, including checkmarked exercises with available answers.', 'The importance of doing homework is emphasized as critical in truly learning linear algebra.', 'Recommendation of a resource for more effective self-study, with a link provided in the description of the video.', 'The chapter cautions against solely relying on watching videos and emphasizes the necessity of completing exercises to learn the material.', 'Flexibility of using the videos as part of a standard course, as a supplement, or for self-study, with the textbook freely available for self-study purposes.', "The chapter recommends 3Blue1Brown's linear algebra videos on YouTube, but underscores the need to complement video resources with homework for effective learning."]}, {'end': 1269.484, 'segs': [{'end': 445.678, 'src': 'embed', 'start': 397.253, 'weight': 0, 'content': [{'end': 401.177, 'text': "And we're only going to solve a particular type of system, what's called a linear system.", 'start': 397.253, 'duration': 3.924}, {'end': 402.878, 'text': 'So I have to introduce those systems.', 'start': 401.237, 'duration': 1.641}, {'end': 406.781, 'text': 'They arise a lot in practice, a real lot in practice.', 'start': 403.258, 'duration': 3.523}, {'end': 408.022, 'text': 'Very important systems.', 'start': 406.881, 'duration': 1.141}, {'end': 414.147, 'text': "But nonetheless, there's only, not completely general, it only applies to certain cases of equations.", 'start': 408.362, 'duration': 5.785}, {'end': 416.109, 'text': 'And so, you know, here we go.', 'start': 414.888, 'duration': 1.221}, {'end': 421.673, 'text': 'So a linear combination of variables, x1 through xn, it looks like this.', 'start': 416.509, 'duration': 5.164}, {'end': 424.816, 'text': 'You have some numbers, a1, a2, a3, etc.', 'start': 421.893, 'duration': 2.923}, {'end': 425.256, 'text': ', through an.', 'start': 424.816, 'duration': 0.44}, {'end': 426.097, 'text': 'Those are real numbers.', 'start': 425.296, 'duration': 0.801}, {'end': 427.037, 'text': '3.4, 7, negative 5, kind of thing.', 'start': 426.117, 'duration': 0.92}, {'end': 436.013, 'text': 'The variables x1, x2, x3, etc.', 'start': 433.052, 'duration': 2.961}, {'end': 440.696, 'text': 'Those letters, x1, x2, those do not have x1 squared.', 'start': 436.974, 'duration': 3.722}, {'end': 442.657, 'text': "It doesn't say the square root of x1.", 'start': 440.956, 'duration': 1.701}, {'end': 444.518, 'text': "It doesn't say arc cotangent of x1.", 'start': 442.757, 'duration': 1.761}, {'end': 445.678, 'text': 'It just says x1.', 'start': 444.678, 'duration': 1}], 'summary': 'Introduction to solving linear systems, important in practice, but applies to specific cases and involves real numbers.', 'duration': 48.425, 'max_score': 397.253, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI397253.jpg'}, {'end': 526.646, 'src': 'embed', 'start': 496.251, 'weight': 8, 'content': [{'end': 497.392, 'text': "I'm calling that the constant.", 'start': 496.251, 'duration': 1.141}, {'end': 503.614, 'text': 'But in any event, you have here that you recognize the linear combination on the left and the letter D on the right.', 'start': 497.452, 'duration': 6.162}, {'end': 512.857, 'text': 'And when we can, we will try to always rearrange the linear equations that we have to have all the variables on one side, always the left,', 'start': 504.574, 'duration': 8.283}, {'end': 519.482, 'text': 'almost always the left and the constant, the number not associated with the variable on the right.', 'start': 512.857, 'duration': 6.625}, {'end': 522.524, 'text': 'And then we always want to solve these things.', 'start': 521.003, 'duration': 1.521}, {'end': 526.646, 'text': 'So we always want to find some real numbers that work in the equation.', 'start': 522.744, 'duration': 3.902}], 'summary': 'Rearranging linear equations to solve for real numbers.', 'duration': 30.395, 'max_score': 496.251, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI496251.jpg'}, {'end': 916.694, 'src': 'embed', 'start': 886.306, 'weight': 2, 'content': [{'end': 889.549, 'text': 'And then knowing that z equals 2 and y, you can solve the first row.', 'start': 886.306, 'duration': 3.243}, {'end': 895.305, 'text': 'So the bottom row clearly says z equals 2.', 'start': 891.75, 'duration': 3.555}, {'end': 899.827, 'text': 'With that, you can figure out what is the value of y.', 'start': 895.305, 'duration': 4.522}, {'end': 905.689, 'text': "y is one and next up because there's only one row that involves an x.", 'start': 899.827, 'duration': 5.862}, {'end': 910.071, 'text': "you'll substitute the y and z stuff into the x row and you end up with what x equals four.", 'start': 905.689, 'duration': 4.382}, {'end': 915.974, 'text': 'Okay, so we now know that the original system had.', 'start': 911.752, 'duration': 4.222}, {'end': 916.694, 'text': 'there we go.', 'start': 915.974, 'duration': 0.72}], 'summary': 'Solving the system, z=2, y=1, x=4.', 'duration': 30.388, 'max_score': 886.306, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI886306.jpg'}, {'end': 1189.069, 'src': 'embed', 'start': 1160.342, 'weight': 4, 'content': [{'end': 1166.043, 'text': 'the goal is to turn this, to use this x, to turn that x into a zero, all right.', 'start': 1160.342, 'duration': 5.701}, {'end': 1167.464, 'text': "so everybody's a smart person.", 'start': 1166.043, 'duration': 1.421}, {'end': 1169.044, 'text': "i don't need to go on a great length.", 'start': 1167.464, 'duration': 1.58}, {'end': 1177.826, 'text': 'so minus one half times two, and add makes zero x minus one half times minus three makes positive 3 halves and add makes 3 halves.', 'start': 1169.044, 'duration': 8.782}, {'end': 1183.208, 'text': 'Minus 1 half times minus 1 makes positive 1 half, and add to 3z makes 7z.', 'start': 1178.307, 'duration': 4.901}, {'end': 1189.069, 'text': 'Minus 1 half times positive 2 makes minus 1w, add to 1w makes 0w.', 'start': 1183.668, 'duration': 5.401}], 'summary': 'Goal: transform x into zero using specific calculations.', 'duration': 28.727, 'max_score': 1160.342, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI1160342.jpg'}], 'start': 335.234, 'title': "Solving linear equations with gauss's method", 'summary': "Emphasizes the importance of doing exercises, particularly the checkmarked ones, to understand the method of solving linear equations using gauss's method, which is a very important mathematical activity. it explains linear combinations, linear equations, and the gauss's method for solving systems of linear equations, emphasizing the importance of finding real number solutions and the systematic approach for transforming the system to obtain the solution, with a focus on efficiency and accuracy. it also introduces gauss's method for solving linear systems, highlighting its effectiveness in providing the right answer and its superiority over other methods, and emphasizes its importance for the semester's course. it also discusses the triangular form and echelon formation as key steps in the method.", 'chapters': [{'end': 416.109, 'start': 335.234, 'title': "Solving linear equations with gauss's method", 'summary': "Emphasizes the importance of doing exercises, particularly the checkmarked ones, to understand the method of solving linear equations using gauss's method, which is a very important mathematical activity.", 'duration': 80.875, 'highlights': ["The importance of doing exercises, particularly the checkmarked ones, to understand the method of solving linear equations using Gauss's method.", 'The chapter emphasizes the significance of solving linear systems, which arise frequently in practice and are very important.', 'Emphasizing the necessity of trying exercises instead of just reading and looking at the answers for effective learning.']}, {'end': 863.576, 'start': 416.509, 'title': "Linear equations and gauss's method", 'summary': "Explains linear combinations, linear equations, and the gauss's method for solving systems of linear equations, emphasizing the importance of finding real number solutions and the systematic approach for transforming the system to obtain the solution, with a focus on efficiency and accuracy.", 'duration': 447.067, 'highlights': ['The chapter introduces linear combinations of variables and coefficients, exemplifying a linear combination of three letters as one quarter x plus y minus z.', 'The concept of linear equations is explained as a linear combination transformed into an equation, with the focus on rearranging equations, finding real number solutions, and defining the solution or n-tuple satisfying the equation.', "The chapter delves into solving systems of linear equations, emphasizing the importance of finding solutions for multiple equations, and introduces Gauss's method as a systematic approach for obtaining the solution with high efficiency and accuracy."]}, {'end': 1269.484, 'start': 864.611, 'title': "Gauss's method for solving linear systems", 'summary': "Introduces gauss's method for solving linear systems, highlighting its effectiveness in providing the right answer and its superiority over other methods, and emphasizes its importance for the semester's course. it also discusses the triangular form and echelon formation as key steps in the method.", 'duration': 404.873, 'highlights': ["Gauss's method provides the right answer and never gives the wrong answer, making it the ideal mathematical procedure for solving linear systems.", "The chapter emphasizes the importance of Gauss's method in the context of the semester's course, indicating that it will be the basis for solving linear systems throughout the course.", 'The process involves transforming the system into a triangular form, where the leading variables are z, y, and x, leading to an easily obtainable solution.']}], 'duration': 934.25, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI335234.jpg', 'highlights': ["The importance of doing exercises, particularly the checkmarked ones, to understand the method of solving linear equations using Gauss's method.", 'The chapter introduces linear combinations of variables and coefficients, exemplifying a linear combination of three letters as one quarter x plus y minus z.', 'Emphasizing the necessity of trying exercises instead of just reading and looking at the answers for effective learning.', 'The chapter emphasizes the significance of solving linear systems, which arise frequently in practice and are very important.', 'The concept of linear equations is explained as a linear combination transformed into an equation, with the focus on rearranging equations, finding real number solutions, and defining the solution or n-tuple satisfying the equation.', "The chapter delves into solving systems of linear equations, emphasizing the importance of finding solutions for multiple equations, and introduces Gauss's method as a systematic approach for obtaining the solution with high efficiency and accuracy.", "Gauss's method provides the right answer and never gives the wrong answer, making it the ideal mathematical procedure for solving linear systems.", "The chapter emphasizes the importance of Gauss's method in the context of the semester's course, indicating that it will be the basis for solving linear systems throughout the course.", 'The process involves transforming the system into a triangular form, where the leading variables are z, y, and x, leading to an easily obtainable solution.']}, {'end': 3272.241, 'segs': [{'end': 2208.401, 'src': 'embed', 'start': 2180.459, 'weight': 3, 'content': [{'end': 2187.064, 'text': "Next up is to use that minus y to get rid of the minus 4y, and you end up with echelon form and you're like okay.", 'start': 2180.459, 'duration': 6.605}, {'end': 2190.587, 'text': "well, no, it doesn't say 0 equals 5 or 0 equals 21,.", 'start': 2187.064, 'duration': 3.523}, {'end': 2197.633, 'text': "so it doesn't have some impossibility, but it also doesn't have an equation for z.", 'start': 2190.587, 'duration': 7.046}, {'end': 2202.496, 'text': "It's got an equation where y leads, it's got an equation where x leads, but it doesn't have an equation where z leads.", 'start': 2197.633, 'duration': 4.863}, {'end': 2208.401, 'text': "So there's no way to get started on that back substitution process.", 'start': 2204.938, 'duration': 3.463}], 'summary': "The echelon form doesn't have an equation for z, hindering the back substitution process.", 'duration': 27.942, 'max_score': 2180.459, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI2180459.jpg'}, {'end': 2587.28, 'src': 'embed', 'start': 2560.802, 'weight': 4, 'content': [{'end': 2567.627, 'text': "And here's the technique, and I can't overemphasize how much we will be doing this technique throughout the course of the semester.", 'start': 2560.802, 'duration': 6.825}, {'end': 2575.092, 'text': "It looks like a small point when you first see it, but it is a key to understanding much of what we're going to do.", 'start': 2567.787, 'duration': 7.305}, {'end': 2579.274, 'text': 'So the idea is to take the letter that is in some sense missing.', 'start': 2576.192, 'duration': 3.082}, {'end': 2583.958, 'text': "Z doesn't have a row, so we're going to take Z.", 'start': 2580.095, 'duration': 3.863}, {'end': 2587.28, 'text': 'You could do other things, but this will turn out to always be the thing we want to do.', 'start': 2583.958, 'duration': 3.322}], 'summary': 'Emphasize the importance of taking the missing letter z for understanding the course material.', 'duration': 26.478, 'max_score': 2560.802, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI2560802.jpg'}, {'end': 2659.586, 'src': 'embed', 'start': 2626.362, 'weight': 8, 'content': [{'end': 2628.365, 'text': "Don't we have a mixture of x and y there?", 'start': 2626.362, 'duration': 2.003}, {'end': 2633.472, 'text': "And the answer, of course, is that we're going to substitute into that first row.", 'start': 2628.665, 'duration': 4.807}, {'end': 2636.336, 'text': "So in the place of y, we're going to write minus 6 plus 4z.", 'start': 2633.532, 'duration': 2.804}, {'end': 2642.839, 'text': "So we get minus x, and there's the minus 6 plus 4z, and there's the 3z.", 'start': 2638.237, 'duration': 4.602}, {'end': 2647.121, 'text': 'And you do the necessary algebra, and you bring everything over to the other side of the equation.', 'start': 2643.259, 'duration': 3.862}, {'end': 2652.003, 'text': "It turns out that when all the smoke clears, you get x equals, and then there's a 3 minus z.", 'start': 2647.181, 'duration': 4.822}, {'end': 2659.586, 'text': 'So I now have z, of course, z is itself, but I now have the other letters, the y and the x, in terms of z.', 'start': 2652.003, 'duration': 7.583}], 'summary': 'Solving equations results in x = 3 - z and other variables in terms of z.', 'duration': 33.224, 'max_score': 2626.362, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI2626362.jpg'}, {'end': 2728.333, 'src': 'embed', 'start': 2697.647, 'weight': 1, 'content': [{'end': 2699.047, 'text': 'however many XYZs you like.', 'start': 2697.647, 'duration': 1.4}, {'end': 2704.05, 'text': "It's obvious that there are infinitely many XYZs, because they have to be different.", 'start': 2699.628, 'duration': 4.422}, {'end': 2707.311, 'text': "It can't be the case that they're all the same, because they all differ, at least in the Z place.", 'start': 2704.07, 'duration': 3.241}, {'end': 2713.494, 'text': 'So there are infinitely many solutions here, and I will say, parameterized by Z.', 'start': 2709.092, 'duration': 4.402}, {'end': 2719.806, 'text': 'It enables you to find solutions to the system, to the starting system,', 'start': 2716.423, 'duration': 3.383}, {'end': 2726.191, 'text': "but it also enables you to conclude something isn't a solution to the starting system, because if somebody says to you what about 1,, 6,", 'start': 2719.806, 'duration': 6.385}, {'end': 2728.333, 'text': '3? You say to yourself 3..', 'start': 2726.191, 'duration': 2.142}], 'summary': 'Infinitely many xyzs, parameterized by z, enable solutions to the system.', 'duration': 30.686, 'max_score': 2697.647, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI2697647.jpg'}, {'end': 2784.929, 'src': 'embed', 'start': 2759.904, 'weight': 0, 'content': [{'end': 2766.694, 'text': 'There are some letters, only one letter in this picture, there are some letters missing from the echelon form system.', 'start': 2759.904, 'duration': 6.79}, {'end': 2768.616, 'text': 'There are some letters that do not lead a row.', 'start': 2766.714, 'duration': 1.902}, {'end': 2771.938, 'text': 'We use those letters to express the other letters.', 'start': 2769.296, 'duration': 2.642}, {'end': 2776.962, 'text': 'We parameterize the leading variables in terms of those that are not leading variables.', 'start': 2772.239, 'duration': 4.723}, {'end': 2784.929, 'text': 'And that parameterization serves as a criteria for what is a solution and what is not a solution.', 'start': 2779.885, 'duration': 5.044}], 'summary': 'Letters in echelon form system serve as criteria for solutions.', 'duration': 25.025, 'max_score': 2759.904, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI2759904.jpg'}], 'start': 1270.503, 'title': "Gauss's method for solving systems", 'summary': "Discusses gauss's method for transforming systems of equations into echelon form and finding solutions, including identifying unique solutions, infinite solutions, and parameterization for generating infinitely many solutions.", 'chapters': [{'end': 1459.691, 'start': 1270.503, 'title': "Gauss's method for solving systems", 'summary': "Discusses using gauss's method to transform a system of equations into echelon form, proceeding through a sequence of steps to find the solution x=1, y=0, z=2, w=-1, which was used by gauss to find the orbit of the first known asteroid.", 'duration': 189.188, 'highlights': ["Using Gauss's method to transform the system into echelon form and finding the solution x=1, y=0, z=2, w=-1.", 'Gauss using this method to find the orbit of the first known asteroid.', 'The process of swapping equations to solve the system.']}, {'end': 2157.21, 'start': 1460.231, 'title': "Gauss's method and system of equations", 'summary': "Discusses gauss's method for solving systems of equations, including the three elementary reduction operations, and explains how it can identify systems with unique solutions or no solutions, using examples and detailed explanations.", 'duration': 696.979, 'highlights': ["Gauss's method involves three operations: multiplying both sides of an equation by a non-zero number, replacing an equation, and row combination, which is the most important step in the process.", "Gauss's method is used in a systematic way to arrange the system of equations, so that the answer is obvious, and it is called Gauss's method, which involves elementary reduction operations.", "Gauss's method can identify systems that don't have a single solution, and it is the method that tells us what the answer is and whether there is no solution.", "The method explains how to construct systems of linear equations to have a unique solution, and how to identify systems that do not have a unique solution using Gauss's method.", "Gauss's method can also identify when there is redundant information in a system of equations, which can lead to no unique solution."]}, {'end': 2443.113, 'start': 2157.21, 'title': "Gauss's method and infinite solutions", 'summary': "Illustrates how to identify infinite solutions in a system of linear equations using gauss's method and echelon form, emphasizing the absence of an equation for a specific variable as a key indicator, resulting in infinitely many solutions for the system.", 'duration': 285.903, 'highlights': ["Identification of infinite solutions in a system of linear equations using Gauss's method and echelon form", 'Absence of an equation for a specific variable as a key indicator of infinitely many solutions', 'Illustration of various values of z leading to infinitely many solutions']}, {'end': 2758.983, 'start': 2444.074, 'title': 'Describing solution sets', 'summary': 'Discusses the technique of describing infinitely many solutions to a linear system through parameterization, enabling the generation of an infinite number of solutions, and provides a criterion for determining solutionhood.', 'duration': 314.909, 'highlights': ['The technique of parameterization enables the generation of an infinite number of solutions.', 'Parameterization provides a criterion for determining solutionhood.', 'The focus on describing infinitely many solutions through parameterization is emphasized as a key technique in the course.']}, {'end': 3272.241, 'start': 2759.904, 'title': 'Parameterization of echelon form linear systems', 'summary': 'Discusses the parameterization of echelon form linear systems, highlighting the process of expressing solutions using free variables and the importance of parameterization. the solutions are described in terms of the free variables, with a focus on the convenience and effectiveness of this approach in determining solutions. the chapter concludes with a preview of the next topic on matrices and vectors.', 'duration': 512.337, 'highlights': ['The solutions are described in terms of the free variables, emphasizing the importance of parameterization.', 'The chapter discusses the convenience and effectiveness of expressing solutions using free variables.', 'Preview of the next topic on matrices and vectors.']}], 'duration': 2001.738, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI1270503.jpg', 'highlights': ["Using Gauss's method to transform the system into echelon form and finding the solution x=1, y=0, z=2, w=-1.", "Identification of infinite solutions in a system of linear equations using Gauss's method and echelon form", 'The technique of parameterization enables the generation of an infinite number of solutions.', "Gauss's method involves three operations: multiplying both sides of an equation by a non-zero number, replacing an equation, and row combination, which is the most important step in the process.", 'The process of swapping equations to solve the system.', "Gauss's method can identify systems that don't have a single solution, and it is the method that tells us what the answer is and whether there is no solution.", "The method explains how to construct systems of linear equations to have a unique solution, and how to identify systems that do not have a unique solution using Gauss's method.", 'Absence of an equation for a specific variable as a key indicator of infinitely many solutions', 'The solutions are described in terms of the free variables, emphasizing the importance of parameterization.', "Gauss's method can also identify when there is redundant information in a system of equations, which can lead to no unique solution."]}, {'end': 4737.743, 'segs': [{'end': 3320.035, 'src': 'embed', 'start': 3294.455, 'weight': 5, 'content': [{'end': 3299.956, 'text': "At the moment, we're not interpreting it so much as the kinds of things you see in a physics class or a Calculus 3 class.", 'start': 3294.455, 'duration': 5.501}, {'end': 3303.996, 'text': "At the moment, we're just going to introduce them as a place to put numbers.", 'start': 3300.436, 'duration': 3.56}, {'end': 3305.157, 'text': "That's all.", 'start': 3304.617, 'duration': 0.54}, {'end': 3307.497, 'text': "But soon enough, we'll give them a more conceptual meaning.", 'start': 3305.317, 'duration': 2.18}, {'end': 3310.218, 'text': 'But at the moment, just simply a place to hold numbers.', 'start': 3307.517, 'duration': 2.701}, {'end': 3320.035, 'text': 'So an M by N matrix, is a rectangular array of numbers, just numbers, no particular physical meaning, m rows, n columns.', 'start': 3310.678, 'duration': 9.357}], 'summary': 'Introducing matrices as a place to put numbers, no conceptual meaning yet.', 'duration': 25.58, 'max_score': 3294.455, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI3294455.jpg'}, {'end': 4626.376, 'src': 'heatmap', 'start': 3356.383, 'weight': 0.987, 'content': [{'end': 3359.144, 'text': 'you often use the associated lowercase Roman letters.', 'start': 3356.383, 'duration': 2.761}, {'end': 3361.065, 'text': "That's the convention that I'm going to be sticking to.", 'start': 3359.164, 'duration': 1.901}, {'end': 3370.507, 'text': "Not 100% of the time, but 95% of the time we'll be writing here capital Roman letter for the matrix and lower Roman letter for the entry.", 'start': 3361.445, 'duration': 9.062}, {'end': 3372.487, 'text': "And you'll often see that in practice, very often.", 'start': 3370.547, 'duration': 1.94}, {'end': 3379.749, 'text': 'Okay, column vector is often just called a vector, is a matrix with a single column.', 'start': 3374.728, 'duration': 5.021}, {'end': 3383.178, 'text': "If a matrix has a single row, it's called a row vector.", 'start': 3380.836, 'duration': 2.342}, {'end': 3388.361, 'text': "And again, you may have seen these things where they have some meaning, and we'll get to those soon enough,", 'start': 3383.218, 'duration': 5.143}, {'end': 3391.784, 'text': "but at the moment we're just simply using them as places to hold numbers.", 'start': 3388.361, 'duration': 3.423}, {'end': 3394.606, 'text': 'The entries of a vector are sometimes called components.', 'start': 3392.164, 'duration': 2.442}, {'end': 3397.008, 'text': 'Instead of entry up here, you often call it component.', 'start': 3394.726, 'duration': 2.282}, {'end': 3402.732, 'text': 'And the column or row vector whose components are all zeros is the zero vector, happens to come up a lot in practice.', 'start': 3397.428, 'duration': 5.304}, {'end': 3408.914, 'text': 'I like to denote vectors with an over arrow, with an arrow on top.', 'start': 3404.813, 'duration': 4.101}, {'end': 3416.536, 'text': 'So let me bring up my, there we go, and bring up my piece of paper here.', 'start': 3409.374, 'duration': 7.162}, {'end': 3419.837, 'text': "And so I'll, whoops, a little shadow there.", 'start': 3417.176, 'duration': 2.661}, {'end': 3425.138, 'text': "So I'll denote vectors, there we go.", 'start': 3420.617, 'duration': 4.521}, {'end': 3428.299, 'text': "I'll denote vectors with an over arrow.", 'start': 3426.478, 'duration': 1.821}, {'end': 3433.24, 'text': "In handwriting, I'll write something like that, V or W.", 'start': 3429.839, 'duration': 3.401}, {'end': 3435.016, 'text': 'Something like that.', 'start': 3434.235, 'duration': 0.781}, {'end': 3439.102, 'text': "Okay? So I'll denote vectors with an over arrow.", 'start': 3436.258, 'duration': 2.844}, {'end': 3441.825, 'text': 'Let me bring it back to the.', 'start': 3439.122, 'duration': 2.703}, {'end': 3442.306, 'text': 'There we go.', 'start': 3441.825, 'duration': 0.481}, {'end': 3447.874, 'text': 'In the textbook, the over arrow is easier to read than my handwriting.', 'start': 3444.389, 'duration': 3.485}, {'end': 3449.215, 'text': "That's just the way it goes.", 'start': 3447.914, 'duration': 1.301}, {'end': 3455.503, 'text': 'This particular column vector has three components, minus one, minus one half, and zero.', 'start': 3449.878, 'duration': 5.625}, {'end': 3463.33, 'text': 'Or I should have mentioned I like over arrow because you often see boldface, very often see boldface in other books,', 'start': 3455.523, 'duration': 7.807}, {'end': 3466.353, 'text': "but I can't do boldface in handwriting or on a blackboard.", 'start': 3463.33, 'duration': 3.023}, {'end': 3469.496, 'text': 'so I like the over arrow and I put it in both places.', 'start': 3466.353, 'duration': 3.143}, {'end': 3472.359, 'text': 'But boldface is very common, very common.', 'start': 3469.857, 'duration': 2.502}, {'end': 3474.581, 'text': 'In fact, maybe more common than over arrow.', 'start': 3472.379, 'duration': 2.202}, {'end': 3481.128, 'text': 'So this is a row vector with three components, and this is the two-component zero vector.', 'start': 3476.721, 'duration': 4.407}, {'end': 3483.792, 'text': 'See a zero with an over arrow.', 'start': 3482.33, 'duration': 1.462}, {'end': 3495.41, 'text': "Oops I'm going to want to be able to add these and to rescale them to double them or triple them, so I need to define those operations.", 'start': 3485.255, 'duration': 10.155}, {'end': 3501.655, 'text': 'So this says if you have two vectors, u and v, then you can add them by adding the components.', 'start': 3495.53, 'duration': 6.125}, {'end': 3507.039, 'text': "You notice, don't you, that this says down to un and this says down to v, and so they are the same size.", 'start': 3501.995, 'duration': 5.044}, {'end': 3510.001, 'text': 'If this one has five entries, then this one has five entries also.', 'start': 3507.139, 'duration': 2.862}, {'end': 3514.285, 'text': 'Anyway, if they have the same number of entries, then you add just by adding the components.', 'start': 3510.682, 'duration': 3.603}, {'end': 3518.313, 'text': 'And you can also rescale the components.', 'start': 3516.112, 'duration': 2.201}, {'end': 3521.234, 'text': "That's called a scalar multiplication of the real number r.", 'start': 3518.333, 'duration': 2.901}, {'end': 3523.255, 'text': 'In this context, r is called a scalar.', 'start': 3521.234, 'duration': 2.021}, {'end': 3529.377, 'text': 'And if, for example, you want to multiply the whole vector by r, why you multiply each of its components by r.', 'start': 3524.195, 'duration': 5.182}, {'end': 3536.603, 'text': "And the advantage of all this is that we can take linear combinations, and that's the key in this course.", 'start': 3531.562, 'duration': 5.041}, {'end': 3542.125, 'text': 'We can take linear combinations of vectors 3 times 1, 2 minus 2 times 0, 1.', 'start': 3536.863, 'duration': 5.262}, {'end': 3546.186, 'text': 'And when you do 3 times 1 minus 2 times 0, you find yourself at 3.', 'start': 3542.125, 'duration': 4.061}, {'end': 3551.047, 'text': 'When you do 3 times 2 minus 2 times 1, you find yourself at 4.', 'start': 3546.186, 'duration': 4.861}, {'end': 3557.249, 'text': 'So these definitions on the top half of the slide enable me to do linear combinations as the example on the bottom half of the slide.', 'start': 3551.047, 'duration': 6.202}, {'end': 3566.664, 'text': "one advantage of the matrix notation is, when you write these systems down, there's a lot of sort of writing x's and y's and equal signs,", 'start': 3560.018, 'duration': 6.646}, {'end': 3574.05, 'text': "and we can get rid of all that just by writing the numbers themselves the minus three, the zero, the two and that's what i have here.", 'start': 3566.664, 'duration': 7.386}, {'end': 3578.434, 'text': "so i can turn this system into what's called here an augmented matrix.", 'start': 3574.05, 'duration': 4.384}, {'end': 3586.659, 'text': 'the vertical bar makes it augmented by just simply writing down the real number coefficients on the left of the vertical bar and the constants on the right of the vertical bar.', 'start': 3578.434, 'duration': 8.225}, {'end': 3590.301, 'text': 'The vertical bar just reminds me that the ones on the left and the ones on the right are a little different.', 'start': 3586.679, 'duration': 3.622}, {'end': 3596.224, 'text': "Do Gauss's method the same Gauss's method steps as we've been doing for so long.", 'start': 3592.162, 'duration': 4.062}, {'end': 3600.326, 'text': 'This is obviously echelon form, and it so happens to be the case.', 'start': 3596.584, 'duration': 3.742}, {'end': 3606.69, 'text': "you see that there is a row with x leading, there is a row with y leading, but there's no row with z leading.", 'start': 3600.326, 'duration': 6.364}, {'end': 3609.291, 'text': "so we know that there's going to be infinitely many solutions.", 'start': 3606.69, 'duration': 2.601}, {'end': 3613.963, 'text': 'And we see the two equations on this.', 'start': 3612.122, 'duration': 1.841}, {'end': 3616.504, 'text': 'This abbreviates the two equations.', 'start': 3614.703, 'duration': 1.801}, {'end': 3619.486, 'text': 'Minus 3x plus 2z equals minus 1.', 'start': 3616.905, 'duration': 2.581}, {'end': 3620.507, 'text': 'There we go for the top.', 'start': 3619.486, 'duration': 1.021}, {'end': 3623.748, 'text': 'And minus 2y plus 8 thirds z equals minus 2.', 'start': 3620.867, 'duration': 2.881}, {'end': 3624.889, 'text': 'There we go for the bottom.', 'start': 3623.748, 'duration': 1.141}, {'end': 3633.088, 'text': "There's no reason why I can't take those two equations and do what I've been doing before, which is to parameterize, that is to say,", 'start': 3627.346, 'duration': 5.742}, {'end': 3637.409, 'text': 'take z and use it as the variable with which you express the other variables.', 'start': 3633.088, 'duration': 4.321}, {'end': 3642.911, 'text': 'So you express x in terms of z and y in terms of z, and you end up with this three equations here.', 'start': 3637.429, 'duration': 5.482}, {'end': 3647.932, 'text': "x is expressed in terms of z, y is expressed in terms of z, and it's a little silly, but z is expressed in terms of z.", 'start': 3643.111, 'duration': 4.821}, {'end': 3654.813, 'text': 'That gives me this solution set, and I wrote it in vector notation here.', 'start': 3651.172, 'duration': 3.641}, {'end': 3662.517, 'text': 'The vector notation is very nice here because it organizes the constants, the one-third, the one, and the zero, all in one place.', 'start': 3655.354, 'duration': 7.163}, {'end': 3667.278, 'text': 'And the coefficients of z, the two-thirds, the four-thirds, and the one, all in one place here.', 'start': 3662.897, 'duration': 4.381}, {'end': 3671.22, 'text': 'So this is an especially organized way to write down the information that is here.', 'start': 3667.318, 'duration': 3.902}, {'end': 3677.877, 'text': "And it's awfully nice because right away we see, for example, what happens when you plug in different z's.", 'start': 3673.413, 'duration': 4.464}, {'end': 3686.303, 'text': 'If you plug in a zero or a one, or a two or a minus, one, half for z, you get the whole family of solutions, the whole family of x, y,', 'start': 3678.217, 'duration': 8.086}, {'end': 3688.305, 'text': "z's written out there very plainly.", 'start': 3686.303, 'duration': 2.002}, {'end': 3691.788, 'text': 'So this is an organized way to write this.', 'start': 3689.086, 'duration': 2.702}, {'end': 3696.332, 'text': 'just to have a second example.', 'start': 3695.031, 'duration': 1.301}, {'end': 3702.077, 'text': "There's nothing special about this example so much, just a simply different example illustrating the same thing.", 'start': 3696.372, 'duration': 5.705}, {'end': 3706.381, 'text': "Here's a system and I can write it in this augmented matrix form.", 'start': 3702.478, 'duration': 3.903}, {'end': 3718.455, 'text': "When you do Gauss's method, there you get echelon form and you convert it to this, to this way of writing.", 'start': 3707.482, 'duration': 10.973}, {'end': 3726.103, 'text': 'the solution set the way of writing the infinitely many different vectors that all solve this system infinitely many different x, y, z,', 'start': 3718.455, 'duration': 7.648}, {'end': 3728.246, 'text': "w's that all solve this system.", 'start': 3726.103, 'duration': 2.143}, {'end': 3735.394, 'text': "I started with you see, there is a z times and a w times, and there's also some constants.", 'start': 3728.246, 'duration': 7.148}, {'end': 3743.922, 'text': 'So again, the advantage of this particular notation is that it shows the constants isolated,', 'start': 3736.618, 'duration': 7.304}, {'end': 3747.324, 'text': 'the coefficients of z isolated and the coefficients of w isolated.', 'start': 3743.922, 'duration': 3.402}, {'end': 3752.327, 'text': "So it's an organized way of doing the job that we've been doing so far already.", 'start': 3747.344, 'duration': 4.983}, {'end': 3760.811, 'text': 'OK So the next section is going to be noticing something about that particular way of writing the solution set.', 'start': 3752.847, 'duration': 7.964}, {'end': 3762.452, 'text': "And we'll see you then next time.", 'start': 3761.211, 'duration': 1.241}, {'end': 3763.012, 'text': 'Very nice.', 'start': 3762.512, 'duration': 0.5}, {'end': 3764.914, 'text': "And that's, there we go.", 'start': 3763.713, 'duration': 1.201}, {'end': 3768.235, 'text': 'Okay, bye-bye.', 'start': 3766.894, 'duration': 1.341}, {'end': 3771.416, 'text': 'Hello, hi, hello.', 'start': 3770.116, 'duration': 1.3}, {'end': 3780.601, 'text': 'Okay, last time we talked about solving a linear system, finding the solution set description using matrix and vector notation.', 'start': 3771.977, 'duration': 8.624}, {'end': 3783.843, 'text': 'And this time I want to give some justification for that.', 'start': 3780.962, 'duration': 2.881}, {'end': 3787.225, 'text': "I want to explain why we're so keen on matrix and vector notation.", 'start': 3783.883, 'duration': 3.342}, {'end': 3792.628, 'text': 'In particular, it enables us to characterize the solution set of the linear system.', 'start': 3787.605, 'duration': 5.023}, {'end': 3794.769, 'text': 'We can completely understand what solutions look like.', 'start': 3792.668, 'duration': 2.101}, {'end': 3799.909, 'text': "Okay, so let's start off by backing up to the previous slide.", 'start': 3795.703, 'duration': 4.206}, {'end': 3802.693, 'text': "That's the slide that we showed in the last video at the end.", 'start': 3799.949, 'duration': 2.744}, {'end': 3805.057, 'text': 'We had a linear system like so.', 'start': 3803.234, 'duration': 1.823}, {'end': 3806.579, 'text': "We did Gauss's method like so.", 'start': 3805.117, 'duration': 1.462}, {'end': 3810.204, 'text': 'We came up with a complete solution set written in vector notation.', 'start': 3806.619, 'duration': 3.585}, {'end': 3813.529, 'text': 'And you notice that it has really two parts.', 'start': 3811.025, 'duration': 2.504}, {'end': 3823.942, 'text': "There's a part of a vector of constants, and you also notice that the remainder of the solution set has something associated with z,", 'start': 3813.709, 'duration': 10.233}, {'end': 3825.645, 'text': 'something associated with w.', 'start': 3823.942, 'duration': 1.703}, {'end': 3828.108, 'text': "So there's vector times letter, vector times letter.", 'start': 3825.645, 'duration': 2.463}, {'end': 3833.012, 'text': 'We want to show that every solution set can be written in that form.', 'start': 3829.588, 'duration': 3.424}, {'end': 3835.235, 'text': 'Particular here, this is called.', 'start': 3833.232, 'duration': 2.003}, {'end': 3837.397, 'text': 'Homogeneous here, this is called.', 'start': 3835.555, 'duration': 1.842}, {'end': 3843.985, 'text': "So there'll be a vector of constants and then also some letter times vector, letter times vector, letter times vector.", 'start': 3837.658, 'duration': 6.327}, {'end': 3851.334, 'text': 'Okay Okay, so I want to describe that looking at the form of the solution set.', 'start': 3845.007, 'duration': 6.327}, {'end': 3855.056, 'text': 'So here is just like the previous one that I just showed you a second ago.', 'start': 3851.674, 'duration': 3.382}, {'end': 3856.477, 'text': 'Here is a system.', 'start': 3855.096, 'duration': 1.381}, {'end': 3860.159, 'text': "Gauss's method gives you some answer, gives you echelon form.", 'start': 3856.877, 'duration': 3.282}, {'end': 3862.821, 'text': 'You take the echelon form, you put like so.', 'start': 3860.44, 'duration': 2.381}, {'end': 3866.063, 'text': 'And now I want to ask you to notice something about those.', 'start': 3863.241, 'duration': 2.822}, {'end': 3870.466, 'text': "If you take Z and W, they can be anything you like, they're free.", 'start': 3867.664, 'duration': 2.802}, {'end': 3881.197, 'text': "If you take Z and W to be zero, well then this part's 0, this part's 0, and you just get the vector of constants the 12 fifths for x,", 'start': 3870.766, 'duration': 10.431}, {'end': 3884.679, 'text': 'the minus 1 fifths for y, the 0 for z and the 0 for w.', 'start': 3881.197, 'duration': 3.482}, {'end': 3890.883, 'text': 'That is to say that this vector here in the solution set is a particular solution of the original system.', 'start': 3884.679, 'duration': 6.204}, {'end': 3897.447, 'text': "That if you plug in 12 fifths for x and minus 1 fifths for y and 0 for z and 0 for w, you'll get 2.", 'start': 3891.443, 'duration': 6.004}, {'end': 3903.752, 'text': "And if you plug in 12 fifths for x and minus 1 fifths for y and 0 for z and 0 for w, you'll get 5.", 'start': 3897.447, 'duration': 6.305}, {'end': 3908.798, 'text': 'So this is a particular, this one here is a particular solution of the linear system.', 'start': 3903.752, 'duration': 5.046}, {'end': 3917.909, 'text': "So we can understand this solution set in some way by understanding the parts, and we're starting off by understanding the part, that very first part,", 'start': 3909.639, 'duration': 8.27}, {'end': 3918.891, 'text': 'the vector of constants.', 'start': 3917.909, 'duration': 0.982}, {'end': 3932.297, 'text': 'Okay?. To understand the rest, to understand the vector times a letter, vector times a letter stuff I want to isolate on a particular kind of,', 'start': 3922.675, 'duration': 9.622}, {'end': 3934.539, 'text': 'on a special kind of linear system.', 'start': 3932.297, 'duration': 2.242}, {'end': 3940.884, 'text': 'I want to take those, the constants on the right-hand side, the constants over here on the right-hand side, the 2 and the 5,', 'start': 3934.699, 'duration': 6.185}, {'end': 3942.946, 'text': 'I want to take them and change them into zeros.', 'start': 3940.884, 'duration': 2.062}, {'end': 3950.278, 'text': 'If you change the 2 and the 5 into zeros, then what you get is a system that is simpler in the following sense.', 'start': 3944.716, 'duration': 5.562}, {'end': 3960.183, 'text': 'I can tell at a glance that it has the solution x equals 0, y equals 0, z equals 0, and x equals 0, y equals 0, z equals 0, w equals 0.', 'start': 3950.499, 'duration': 9.684}, {'end': 3962.884, 'text': 'So the vector of zeros definitely solves this system.', 'start': 3960.183, 'duration': 2.701}, {'end': 3967.048, 'text': 'So I know one of the particular solutions of this system.', 'start': 3963.524, 'duration': 3.524}, {'end': 3968.29, 'text': 'It is the vector of zeros.', 'start': 3967.148, 'duration': 1.142}, {'end': 3977.54, 'text': "Gauss's method goes through the exact same steps, and you end up with a solution set that I didn't bother to write the vector of zeros.", 'start': 3968.71, 'duration': 8.83}, {'end': 3981.285, 'text': 'So I have vector times a letter, vector times a letter.', 'start': 3978.782, 'duration': 2.503}, {'end': 3986.711, 'text': 'So this is the other half of the general solution.', 'start': 3983.47, 'duration': 3.241}, {'end': 3994.613, 'text': 'Back here I had a general solution that had a vector with some constants in it and then vector times a letter, vector times a letter.', 'start': 3986.931, 'duration': 7.682}, {'end': 4001.475, 'text': "Well here I've managed to isolate on the vector times a letter, vector times a letter stuff by turning those into zeros.", 'start': 3994.673, 'duration': 6.802}, {'end': 4009.715, 'text': 'So, in general, when you look at a linear equation like the top line or the bottom line here, that has a zero on the right hand side,', 'start': 4002.813, 'duration': 6.902}, {'end': 4013.595, 'text': 'you say that a linear equation is homogeneous if it has a constant of zero.', 'start': 4009.715, 'duration': 3.88}, {'end': 4015.776, 'text': 'So it has a zero on that right hand side.', 'start': 4013.655, 'duration': 2.121}, {'end': 4022.977, 'text': 'Homogeneous And of course the system is homogeneous if it has all zeros on the right hand side.', 'start': 4015.996, 'duration': 6.981}, {'end': 4025.798, 'text': 'Now homogeneous systems are linear systems.', 'start': 4023.978, 'duration': 1.82}, {'end': 4030.139, 'text': 'So just like other linear systems, except they have one special property.', 'start': 4026.198, 'duration': 3.941}, {'end': 4035.16, 'text': 'The homogeneous system can have infinitely many solutions.', 'start': 4031.613, 'duration': 3.547}, {'end': 4037.585, 'text': 'Look at this one here, has infinitely many solutions.', 'start': 4035.2, 'duration': 2.385}, {'end': 4040.812, 'text': 'Homogeneous system can have a unique solution.', 'start': 4038.828, 'duration': 1.984}, {'end': 4042.175, 'text': 'This one here has a unique solution.', 'start': 4040.852, 'duration': 1.323}, {'end': 4048.799, 'text': 'But we know one of the solutions for a homogeneous system is all zeros.', 'start': 4043.516, 'duration': 5.283}, {'end': 4054.083, 'text': 'For example, if you plug x equals zero, y equals zero, z equals zero, you definitely get one of the infinitely many.', 'start': 4049.2, 'duration': 4.883}, {'end': 4058.626, 'text': 'If you plug x equals zero, y equals zero, z equals zero, you get the one solution here.', 'start': 4054.463, 'duration': 4.163}, {'end': 4066.351, 'text': 'So that is to say that homogeneous systems are linear systems, but they cannot have the no solutions possibility.', 'start': 4059.306, 'duration': 7.045}, {'end': 4068.792, 'text': "That can't happen with a homogeneous system.", 'start': 4066.831, 'duration': 1.961}, {'end': 4070.894, 'text': 'Oops, I spelled solution wrong.', 'start': 4069.693, 'duration': 1.201}, {'end': 4080.144, 'text': 'Oops Okay, so in general, if you have a homogeneous system, then the solution set has to look like this.', 'start': 4071.074, 'duration': 9.07}, {'end': 4084.507, 'text': "Vector times a letter, vector times a letter, vector times a letter, however many you've got.", 'start': 4080.484, 'duration': 4.023}, {'end': 4088.57, 'text': 'How many will you have? It says k is the number of free variables.', 'start': 4085.728, 'duration': 2.842}, {'end': 4091.933, 'text': 'So when we had z and w for free variables, why we had two vectors.', 'start': 4088.61, 'duration': 3.323}, {'end': 4098.314, 'text': "Now, I'm going to do something here that we do sometimes.", 'start': 4093.993, 'duration': 4.321}, {'end': 4103.316, 'text': 'Rather than give a proof for the lemma, the proof is in the book but rather than give a proof for the lemma,', 'start': 4099.055, 'duration': 4.261}, {'end': 4107.018, 'text': "we're going to try to convey the idea here of the proof by giving an example.", 'start': 4103.316, 'duration': 3.702}, {'end': 4108.997, 'text': 'Of course, you should refer to the book for the proof.', 'start': 4107.037, 'duration': 1.96}, {'end': 4110.019, 'text': "That's the right thing to do.", 'start': 4109.058, 'duration': 0.961}, {'end': 4119.261, 'text': "So I want to instead convey the idea of the proof since the slides are intended to introduce the discussion that's in the book.", 'start': 4110.959, 'duration': 8.302}, {'end': 4122.368, 'text': 'Okay, so for the main idea.', 'start': 4121.127, 'duration': 1.241}, {'end': 4130.691, 'text': 'you consider this system of homogeneous equations and using the bottom equation you get, you can express y in terms of z and w.', 'start': 4122.368, 'duration': 8.323}, {'end': 4135.832, 'text': "Okay, that's fine, but notice that when you do, there's no constants or a constant of zero, whichever way you like to say it.", 'start': 4130.691, 'duration': 5.141}, {'end': 4144.697, 'text': 'If you take the value of y and you substitute into the top equation again no constants or a constant of zero, whatever way you like to say it,', 'start': 4137.194, 'duration': 7.503}, {'end': 4147.858, 'text': 'and so you get x expressed in terms of z and w with no constants.', 'start': 4144.697, 'duration': 3.161}, {'end': 4151.457, 'text': 'So you end up with no constants or constant of zero.', 'start': 4149.076, 'duration': 2.381}, {'end': 4161.064, 'text': 'So you end up with z times a vector, w times a vector, no vector of constants, or again, a particular solution of the zero vector.', 'start': 4152.358, 'duration': 8.706}, {'end': 4171.152, 'text': "And then this right here, minus two, one, one, zero, that's the beta one, and zero minus one, zero, one, that's the beta k, and the k is two.", 'start': 4162.486, 'duration': 8.666}, {'end': 4178.017, 'text': 'So we have that the general solution can be written as a vector times a letter, vector times a letter, vector times a letter.', 'start': 4171.933, 'duration': 6.084}, {'end': 4185.273, 'text': 'Okay, so the homogeneous solution set is special in some way that we have a characterization of how you can write it,', 'start': 4179.571, 'duration': 5.702}, {'end': 4192.455, 'text': 'and that stems from the fact that when you have a homogeneous system, the particular solution is there is a particular solution, that is,', 'start': 4185.273, 'duration': 7.182}, {'end': 4193.136, 'text': 'the zero vector.', 'start': 4192.455, 'duration': 0.681}, {'end': 4203.899, 'text': 'Okay, so in general, then, for a linear system in any particular solution, the solution set looks of this form P plus H.', 'start': 4195.697, 'duration': 8.202}, {'end': 4208.341, 'text': 'you can, if you give me a linear system, I can write its solution set in this form', 'start': 4203.899, 'duration': 4.442}, {'end': 4214.168, 'text': 'where H satisfies the associated homogeneous system and P is any particular solution.', 'start': 4209.525, 'duration': 4.643}, {'end': 4216.489, 'text': 'Now this one we are going to prove.', 'start': 4215.268, 'duration': 1.221}, {'end': 4217.71, 'text': 'I am going to go through the argument.', 'start': 4216.549, 'duration': 1.161}, {'end': 4220.792, 'text': "It's not very long, but it involves two steps.", 'start': 4217.75, 'duration': 3.042}, {'end': 4225.775, 'text': 'This says that the set of solutions can be written in this form.', 'start': 4221.272, 'duration': 4.503}, {'end': 4227.856, 'text': 'Set of solutions equals this set here.', 'start': 4225.875, 'duration': 1.981}, {'end': 4230.18, 'text': 'where P is any particular solution.', 'start': 4228.479, 'duration': 1.701}, {'end': 4233.943, 'text': 'So this is two sets, a set of solutions and this set here.', 'start': 4230.941, 'duration': 3.002}, {'end': 4235.944, 'text': 'To show that two sets are equal.', 'start': 4234.443, 'duration': 1.501}, {'end': 4238.025, 'text': 'the most straightforward thing to do not the only thing to do,', 'start': 4235.944, 'duration': 2.081}, {'end': 4242.728, 'text': 'but the most straightforward thing to do is to show that each one is a subset of the other.', 'start': 4238.025, 'duration': 4.703}, {'end': 4244.149, 'text': "And that's what we're going to do here.", 'start': 4242.788, 'duration': 1.361}, {'end': 4247.711, 'text': "To give the proof, I couldn't fit it all in one slide, so it's on the second slide too.", 'start': 4244.349, 'duration': 3.362}, {'end': 4255.917, 'text': "But to give the proof, I'm going to show, first of all, that if a vector is in the solution set, then it is in this set also.", 'start': 4248.052, 'duration': 7.865}, {'end': 4265.651, 'text': "And then on the second slide, I'm going to show that if a vector is in the set given, the P plus H set, then that vector solves the system.", 'start': 4257.284, 'duration': 8.367}, {'end': 4267.993, 'text': "So that's called mutual inclusion.", 'start': 4266.491, 'duration': 1.502}, {'end': 4270.174, 'text': "I'm going to show set inclusion both directions.", 'start': 4268.033, 'duration': 2.141}, {'end': 4274.418, 'text': "If a vector solves the system, then it's of this form.", 'start': 4270.875, 'duration': 3.543}, {'end': 4280.042, 'text': 'Okay, so to show that, what do I have to do? Well, you assume that S solves the system.', 'start': 4276.079, 'duration': 3.963}, {'end': 4280.783, 'text': 'So then.', 'start': 4280.403, 'duration': 0.38}, {'end': 4286.631, 'text': 'I want to show that S minus P is the right form for H.', 'start': 4282.87, 'duration': 3.761}, {'end': 4289.932, 'text': 'If I show that S minus P equals H, why then S equals P plus H??', 'start': 4286.631, 'duration': 3.301}, {'end': 4291.672, 'text': 'Okay, so what is H??', 'start': 4290.552, 'duration': 1.12}, {'end': 4298.514, 'text': 'H means that when you plug the components of the vector in, then you got to get zero.', 'start': 4292.473, 'duration': 6.041}, {'end': 4301.955, 'text': "That's what homogeneous system means.", 'start': 4299.114, 'duration': 2.841}, {'end': 4307.719, 'text': "Okay, so I'm taking the components of S minus P, and plugging them into each equation.", 'start': 4302.695, 'duration': 5.024}, {'end': 4312.883, 'text': "I'm only going to show one equation, of course, but I'm taking the components of s minus p and plugging them into that one equation.", 'start': 4307.799, 'duration': 5.084}, {'end': 4316.505, 'text': "Well, you do some algebra here, so let's see.", 'start': 4314.384, 'duration': 2.121}, {'end': 4322.229, 'text': "You distribute the a's, and you get a's times s's minus a's times p's.", 'start': 4316.525, 'duration': 5.704}, {'end': 4326.553, 'text': "Well, A's times S's gives you D.", 'start': 4324.011, 'duration': 2.542}, {'end': 4330.817, 'text': "That's exactly what, S is a solution of the system, so that's exactly what that means.", 'start': 4326.553, 'duration': 4.264}, {'end': 4332.018, 'text': "A's times S's.", 'start': 4330.837, 'duration': 1.181}, {'end': 4337.162, 'text': "A's times the components of S gives you D, and P is a particular solution also of the system.", 'start': 4332.018, 'duration': 5.144}, {'end': 4341.225, 'text': "so A's times P's gives you D and their D must be zero.", 'start': 4337.162, 'duration': 4.063}, {'end': 4343.006, 'text': "So that's exactly what I needed to show.", 'start': 4341.325, 'duration': 1.681}, {'end': 4347.51, 'text': 'I needed to show that S minus P has the right form for H.', 'start': 4343.407, 'duration': 4.103}, {'end': 4353.475, 'text': 'H is a vector that has the property that when you plug it into the left-hand side of the linear system, it will give you zeros.', 'start': 4347.51, 'duration': 5.965}, {'end': 4356.672, 'text': "Okay, so that's half of the argument.", 'start': 4355.27, 'duration': 1.402}, {'end': 4358.155, 'text': 'I have to show the other half of the argument.', 'start': 4356.713, 'duration': 1.442}, {'end': 4368.136, 'text': 'The other half is that if a vector has this form particular solution plus a member of the set of homogeneous solutions,', 'start': 4358.175, 'duration': 9.961}, {'end': 4371.057, 'text': 'then it is a solution of the system that you started with.', 'start': 4368.136, 'duration': 2.921}, {'end': 4379.839, 'text': 'So you take a vector of the form P plus H, and you want to show that P plus H solves the given system.', 'start': 4371.137, 'duration': 8.702}, {'end': 4386.04, 'text': "So that's just a matter of plugging the components of the vector P plus H into each equation.", 'start': 4379.919, 'duration': 6.121}, {'end': 4392.382, 'text': 'So here I plug the components of the vector P plus H, the first component, all the way down to the nth component of the vector P plus H.', 'start': 4386.08, 'duration': 6.302}, {'end': 4395.577, 'text': 'I plugged it into the equation.', 'start': 4393.075, 'duration': 2.502}, {'end': 4396.958, 'text': 'I did the obvious algebra.', 'start': 4395.677, 'duration': 1.281}, {'end': 4398.159, 'text': "I distributed the A's.", 'start': 4397.038, 'duration': 1.121}, {'end': 4402.301, 'text': "So I have the A's times the P's and the A's times the H's.", 'start': 4398.719, 'duration': 3.582}, {'end': 4403.682, 'text': "A's times the P's.", 'start': 4402.862, 'duration': 0.82}, {'end': 4406.004, 'text': 'P is a particular solution of the linear system.', 'start': 4403.742, 'duration': 2.262}, {'end': 4409.526, 'text': "So A times the P's gives you D.", 'start': 4406.084, 'duration': 3.442}, {'end': 4411.948, 'text': 'H on the other hand is a solution of a homogeneous system.', 'start': 4409.526, 'duration': 2.422}, {'end': 4414.73, 'text': 'And I know we only just saw the definition a few minutes ago.', 'start': 4412.268, 'duration': 2.462}, {'end': 4418.112, 'text': 'But the homogeneous system is the one with zeros on the right hand side.', 'start': 4414.99, 'duration': 3.122}, {'end': 4420.033, 'text': "So A's times the H's gives you zero.", 'start': 4418.152, 'duration': 1.881}, {'end': 4422.095, 'text': 'And D plus zero gives you D.', 'start': 4420.994, 'duration': 1.101}, {'end': 4430.351, 'text': 'So in general here, we can always write the solution set in this form.', 'start': 4425.547, 'duration': 4.804}, {'end': 4435.316, 'text': 'We can always write the solution set in the form we started out the day by looking at here.', 'start': 4430.852, 'duration': 4.464}, {'end': 4443.364, 'text': 'You have particular solution, and here you have the homogeneous solution, vector times a letter, vector times a letter.', 'start': 4435.716, 'duration': 7.648}, {'end': 4451.069, 'text': 'Now, we like this for a number of reasons.', 'start': 4448.567, 'duration': 2.502}, {'end': 4458.935, 'text': 'One is that I can write down the solution of a linear system and understand completely what the solution set of a linear system looks like.', 'start': 4452.23, 'duration': 6.705}, {'end': 4464.86, 'text': 'It looks like any particular solution, any particular solution at all that you find can go there.', 'start': 4459.436, 'duration': 5.424}, {'end': 4469.644, 'text': 'And then here is vector times a letter, vector times a letter, vector times a letter, et cetera.', 'start': 4465.34, 'duration': 4.304}, {'end': 4472.926, 'text': 'I just copied that from the solution of the homogeneous.', 'start': 4470.584, 'duration': 2.342}, {'end': 4473.827, 'text': "That's the H part.", 'start': 4472.966, 'duration': 0.861}, {'end': 4482.639, 'text': "And in particular what happens here is that by looking at the previous slide, there's the previous slide.", 'start': 4476.795, 'duration': 5.844}, {'end': 4486.982, 'text': 'by looking at the previous slide and understanding what the various parts can be,', 'start': 4482.639, 'duration': 4.343}, {'end': 4491.005, 'text': 'I can see that solution sets of linear systems have three possibilities.', 'start': 4486.982, 'duration': 4.023}, {'end': 4499.271, 'text': "Solution sets of linear systems are either empty, have one element, that is to say there's a unique solution, or else have infinitely many elements.", 'start': 4491.486, 'duration': 7.785}, {'end': 4501.593, 'text': 'Those are the three possibilities that can happen.', 'start': 4499.831, 'duration': 1.762}, {'end': 4504.495, 'text': "Here I've got a nice box on the next slide that I'm going to point to.", 'start': 4501.633, 'duration': 2.862}, {'end': 4509.771, 'text': 'So in this box, this little array here, there are two possibilities.', 'start': 4506.53, 'duration': 3.241}, {'end': 4512.872, 'text': "Either a particular solution exists or else it doesn't.", 'start': 4509.891, 'duration': 2.981}, {'end': 4515.553, 'text': "And we've certainly seen linear systems that don't have a solution.", 'start': 4512.972, 'duration': 2.581}, {'end': 4517.114, 'text': 'So that would be the bottom line.', 'start': 4515.873, 'duration': 1.241}, {'end': 4520.455, 'text': "Well, if a linear system doesn't have a solution, then there's just no solution.", 'start': 4517.454, 'duration': 3.001}, {'end': 4521.316, 'text': "That's all there is to it.", 'start': 4520.475, 'duration': 0.841}, {'end': 4528.358, 'text': 'On the other hand, if a linear system does have a solution, if a linear system does, oops, I went too far.', 'start': 4523.096, 'duration': 5.262}, {'end': 4535.601, 'text': 'If a linear system does have solution, if there is a P that you could write there, then what can happen for the homogeneous case?', 'start': 4528.638, 'duration': 6.963}, {'end': 4538.172, 'text': 'What can happen with homogeneous systems?', 'start': 4536.331, 'duration': 1.841}, {'end': 4546.177, 'text': "And you'll remember from a couple slides ago homogeneous systems are just like regular systems, except there's one thing that can't happen.", 'start': 4538.953, 'duration': 7.224}, {'end': 4548.398, 'text': 'So here we go in the box.', 'start': 4546.997, 'duration': 1.401}, {'end': 4550.98, 'text': 'Homogeneous systems can have infinitely many.', 'start': 4548.799, 'duration': 2.181}, {'end': 4556.643, 'text': "Homogeneous systems can have unique, but they can't have no solutions.", 'start': 4551.34, 'duration': 5.303}, {'end': 4559.245, 'text': 'So we get this array here.', 'start': 4557.564, 'duration': 1.681}, {'end': 4564.108, 'text': 'So the number of solutions to the homogeneous system can be one or can be infinitely many.', 'start': 4559.525, 'duration': 4.583}, {'end': 4570.728, 'text': 'And if the number of solutions to the homogeneous system is one and a particular solution exists, why, then,', 'start': 4564.847, 'duration': 5.881}, {'end': 4573.169, 'text': 'the system you started with has a unique solution?', 'start': 4570.728, 'duration': 2.441}, {'end': 4579.05, 'text': 'If the number of solutions to the homogeneous system is infinitely many and a particular solution exists,', 'start': 4574.109, 'duration': 4.941}, {'end': 4581.831, 'text': 'then the system you started with has infinitely many solutions.', 'start': 4579.05, 'duration': 2.781}, {'end': 4585.352, 'text': "And of course, the no solutions case means that there's no particular solution.", 'start': 4582.271, 'duration': 3.081}, {'end': 4592.198, 'text': 'Okay, and just to close, let me sneak a word in here before I let you go.', 'start': 4587.375, 'duration': 4.823}, {'end': 4596.781, 'text': 'So an important special case is when the linear system has the same number of equations as unknowns.', 'start': 4592.618, 'duration': 4.163}, {'end': 4603.506, 'text': "So when you do, when you look at the matrix of coefficients for that linear system, there's two cases.", 'start': 4598.422, 'duration': 5.084}, {'end': 4606.968, 'text': "Either it has a unique solution or else it doesn't.", 'start': 4604.006, 'duration': 2.962}, {'end': 4619.03, 'text': 'So in the case where the matrix of coefficients of the homogeneous system has a unique solution, you say that that matrix is non-singular.', 'start': 4610.622, 'duration': 8.408}, {'end': 4626.376, 'text': "Non-singular here means something like normal or usual or doesn't stand out or not particularly problematic.", 'start': 4619.77, 'duration': 6.606}], 'summary': 'The transcript explains matrices, vectors, linear combinations, and the characterization of solution sets using matrix and vector notation, emphasizing the importance of homogeneous systems and the possibilities of unique, infinitely many, or no solutions for linear systems.', 'duration': 1269.993, 'max_score': 3356.383, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI3356383.jpg'}, {'end': 4040.812, 'src': 'embed', 'start': 4015.996, 'weight': 7, 'content': [{'end': 4022.977, 'text': 'Homogeneous And of course the system is homogeneous if it has all zeros on the right hand side.', 'start': 4015.996, 'duration': 6.981}, {'end': 4025.798, 'text': 'Now homogeneous systems are linear systems.', 'start': 4023.978, 'duration': 1.82}, {'end': 4030.139, 'text': 'So just like other linear systems, except they have one special property.', 'start': 4026.198, 'duration': 3.941}, {'end': 4035.16, 'text': 'The homogeneous system can have infinitely many solutions.', 'start': 4031.613, 'duration': 3.547}, {'end': 4037.585, 'text': 'Look at this one here, has infinitely many solutions.', 'start': 4035.2, 'duration': 2.385}, {'end': 4040.812, 'text': 'Homogeneous system can have a unique solution.', 'start': 4038.828, 'duration': 1.984}], 'summary': 'Homogeneous systems can have infinitely many solutions and a unique solution.', 'duration': 24.816, 'max_score': 4015.996, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI4015996.jpg'}, {'end': 4701.849, 'src': 'embed', 'start': 4676.743, 'weight': 0, 'content': [{'end': 4683.645, 'text': "If you're like reading a Sherlock Holmes novel and it says you know the singular case of the dancing men, Singular means stands out or peculiar,", 'start': 4676.743, 'duration': 6.902}, {'end': 4685.505, 'text': 'or interesting, or somewhat unusual.', 'start': 4683.645, 'duration': 1.86}, {'end': 4688.086, 'text': "And that's what's intended to happen here.", 'start': 4685.825, 'duration': 2.261}, {'end': 4694.187, 'text': 'Non-singular case is the case where you find that the linear system has exactly one solution.', 'start': 4688.426, 'duration': 5.761}, {'end': 4696.507, 'text': 'The homogeneous system has exactly one solution.', 'start': 4694.227, 'duration': 2.28}, {'end': 4699.368, 'text': "That's kind of the expected or the usual or the typical.", 'start': 4696.727, 'duration': 2.641}, {'end': 4701.849, 'text': 'Okay, very good.', 'start': 4701.128, 'duration': 0.721}], 'summary': 'Understanding the concept of singular and non-singular cases in linear systems.', 'duration': 25.106, 'max_score': 4676.743, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI4676743.jpg'}], 'start': 3272.261, 'title': 'Matrices and vectors in linear systems', 'summary': 'Introduces matrices and vectors as a new way of writing linear systems, discusses their basics including representation and operations, and characterizes the solution set of a linear system using matrix and vector notation, explaining homogeneous systems and their solution sets.', 'chapters': [{'end': 3310.218, 'start': 3272.261, 'title': 'Introduction to matrices and vectors', 'summary': 'Introduces a new way of writing linear systems and solutions, which simplifies bookkeeping and provides conceptual meaning, initially treating matrices and vectors as placeholders for numbers before assigning them a conceptual meaning.', 'duration': 37.957, 'highlights': ['The chapter introduces a new way of writing linear systems and solutions, simplifying bookkeeping and providing conceptual meaning.', 'The initial interpretation of matrices and vectors is as placeholders for numbers, with plans to assign them a more conceptual meaning later.', 'Matrices and vectors are presented as a place to put numbers, with the intention of giving them a more conceptual meaning in the future.']}, {'end': 3752.327, 'start': 3310.678, 'title': 'Matrices and vectors basics', 'summary': 'Discusses the basics of matrices and vectors, including the representation, notation, and operations such as addition, scalar multiplication, and linear combinations, culminating in the use of matrix notation to solve systems of linear equations and parameterize solutions.', 'duration': 441.649, 'highlights': ['Matrices and Vectors Representation', 'Operations on Vectors and Linear Combinations', 'Matrix Notation for Solving Systems of Linear Equations']}, {'end': 4084.507, 'start': 3752.847, 'title': 'Understanding linear system solution set', 'summary': 'Discusses the use of matrix and vector notation to characterize the solution set of a linear system, showing how every solution set can be written in a particular form and explaining the concept of homogeneous systems with infinitely many solutions and the unique solution.', 'duration': 331.66, 'highlights': ['The chapter discusses the use of matrix and vector notation to characterize the solution set of a linear system, enabling a complete understanding of what solutions look like.', 'It explains how every solution set can be written in a particular form with a vector of constants and some letter times vector, and highlights the concept of homogeneous systems with infinitely many solutions and the unique solution.', 'It emphasizes the special property of homogeneous systems, which can have infinitely many solutions and cannot have the possibility of having no solutions.']}, {'end': 4737.743, 'start': 4085.728, 'title': 'Homogeneous systems and solution sets', 'summary': 'Discusses the proof of representing solution sets of linear systems in the form p+h, where h satisfies the associated homogeneous system and p is any particular solution, highlighting the characterization of homogeneous solution sets and the three possibilities for solution sets of linear systems: empty, unique, or infinitely many solutions.', 'duration': 652.015, 'highlights': ['The chapter discusses the proof of representing solution sets of linear systems in the form P+H, where H satisfies the associated homogeneous system and P is any particular solution, which provides an understanding of the solution set of a linear system.', 'It highlights the three possibilities for solution sets of linear systems: empty, unique, or infinitely many solutions, based on the existence and nature of particular and homogeneous solutions.', 'It explains the concept of non-singular and singular matrices in the context of linear systems, where non-singular matrices are associated with unique solutions and singular matrices are associated with non-unique solutions.']}], 'duration': 1465.482, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI3272261.jpg', 'highlights': ['The chapter introduces a new way of writing linear systems and solutions, simplifying bookkeeping and providing conceptual meaning.', 'The chapter discusses the proof of representing solution sets of linear systems in the form P+H, where H satisfies the associated homogeneous system and P is any particular solution, which provides an understanding of the solution set of a linear system.', 'The chapter discusses the use of matrix and vector notation to characterize the solution set of a linear system, enabling a complete understanding of what solutions look like.', 'It emphasizes the special property of homogeneous systems, which can have infinitely many solutions and cannot have the possibility of having no solutions.', 'It explains the concept of non-singular and singular matrices in the context of linear systems, where non-singular matrices are associated with unique solutions and singular matrices are associated with non-unique solutions.', 'Operations on Vectors and Linear Combinations', 'Matrix Notation for Solving Systems of Linear Equations', 'Matrices and Vectors Representation', 'Matrices and vectors are presented as a place to put numbers, with the intention of giving them a more conceptual meaning in the future.', 'The initial interpretation of matrices and vectors is as placeholders for numbers, with plans to assign them a more conceptual meaning later.']}, {'end': 5992.415, 'segs': [{'end': 4851.348, 'src': 'embed', 'start': 4818.804, 'weight': 2, 'content': [{'end': 4820.786, 'text': "Let's see if I can manipulate my stuff.", 'start': 4818.804, 'duration': 1.982}, {'end': 4821.286, 'text': 'There we go.', 'start': 4820.826, 'duration': 0.46}, {'end': 4822.127, 'text': 'There we go.', 'start': 4821.306, 'duration': 0.821}, {'end': 4823.388, 'text': "And here's the second.", 'start': 4822.527, 'duration': 0.861}, {'end': 4829.352, 'text': 'It just says produce three equations, three unknown systems that have infinitely many solutions.', 'start': 4823.448, 'duration': 5.904}, {'end': 4832.895, 'text': 'Produce one with no solutions and then also produce one with one solution.', 'start': 4829.492, 'duration': 3.403}, {'end': 4837.559, 'text': "So I've got to produce that on a piece of paper here if I can figure out how to do this.", 'start': 4833.395, 'duration': 4.164}, {'end': 4839.12, 'text': 'There we go.', 'start': 4838.64, 'duration': 0.48}, {'end': 4841.042, 'text': 'Produce that on a piece of paper.', 'start': 4839.64, 'duration': 1.402}, {'end': 4842.423, 'text': "So I'm just going to make one up.", 'start': 4841.162, 'duration': 1.261}, {'end': 4844.024, 'text': 'Of course, you would make up a different one.', 'start': 4842.463, 'duration': 1.561}, {'end': 4851.348, 'text': 'The idea with these systems is to produce, for example, infinitely many solutions.', 'start': 4845.424, 'duration': 5.924}], 'summary': 'Producing systems of equations with different solutions, including infinitely many solutions.', 'duration': 32.544, 'max_score': 4818.804, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI4818804.jpg'}, {'end': 5235.954, 'src': 'embed', 'start': 5210.267, 'weight': 0, 'content': [{'end': 5215.651, 'text': 'In this case, for example, you might be going to the left by one and down by one.', 'start': 5210.267, 'duration': 5.384}, {'end': 5219.093, 'text': "That's a displacement, and I've drawn that displacement as an arrow.", 'start': 5215.791, 'duration': 3.302}, {'end': 5223.716, 'text': 'The arrowhead, of course, points to where you end, and the beginning of the arrow points to where you started.', 'start': 5219.373, 'duration': 4.343}, {'end': 5231.471, 'text': 'vectors that say, for example, displacements with the same magnitude in the same direction are equal.', 'start': 5225.506, 'duration': 5.965}, {'end': 5235.954, 'text': 'so i have a bunch of vectors drawn there on the plane and they all have the same magnitude.', 'start': 5231.471, 'duration': 4.483}], 'summary': 'Vectors represent displacements with equal magnitude and direction.', 'duration': 25.687, 'max_score': 5210.267, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI5210267.jpg'}, {'end': 5383.883, 'src': 'embed', 'start': 5355.425, 'weight': 1, 'content': [{'end': 5357.246, 'text': 'Pointing in the same direction, but three times as long.', 'start': 5355.425, 'duration': 1.821}, {'end': 5362.393, 'text': 'If you multiply a vector by minus 1, you make it point in the other direction.', 'start': 5358.112, 'duration': 4.281}, {'end': 5368.455, 'text': "If you multiplied a vector by minus 3, you'd make it point in the other direction and be 3 times as long as it was before.", 'start': 5362.673, 'duration': 5.782}, {'end': 5373.617, 'text': 'So scalars, the 3 or the minus 1 here, rescale the vector.', 'start': 5369.376, 'duration': 4.241}, {'end': 5376.718, 'text': 'The geometry is they simply make it bigger or smaller.', 'start': 5373.877, 'duration': 2.841}, {'end': 5377.178, 'text': "That's all.", 'start': 5376.878, 'duration': 0.3}, {'end': 5381.823, 'text': 'The addition of two vectors, again, familiar.', 'start': 5379.682, 'duration': 2.141}, {'end': 5383.883, 'text': 'I expect that this is a review for most folks.', 'start': 5381.843, 'duration': 2.04}], 'summary': 'Multiplying a vector by minus 3 makes it point in the other direction and be 3 times as long.', 'duration': 28.458, 'max_score': 5355.425, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI5355425.jpg'}, {'end': 5839.818, 'src': 'embed', 'start': 5814.152, 'weight': 3, 'content': [{'end': 5818.993, 'text': "So that's why we're interested, and that's why we call the course linear algebra, because it's about these linear things.", 'start': 5814.152, 'duration': 4.841}, {'end': 5820.835, 'text': 'What are the possibilities?', 'start': 5819.775, 'duration': 1.06}, {'end': 5823.676, 'text': 'question two what are the possibilities for the intersection of two planes?', 'start': 5820.835, 'duration': 2.841}, {'end': 5829.597, 'text': 'So, really, the point of this question is to have a person think about in your head.', 'start': 5824.436, 'duration': 5.161}, {'end': 5836.118, 'text': 'you know, you imagine two planes people, some people are better at imagining these kinds of things than others.', 'start': 5829.597, 'duration': 6.521}, {'end': 5839.818, 'text': "but still, two planes, it's like two pieces of paper.", 'start': 5836.118, 'duration': 3.7}], 'summary': 'Linear algebra course covers intersection of two planes visually.', 'duration': 25.666, 'max_score': 5814.152, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI5814152.jpg'}, {'end': 5940.146, 'src': 'embed', 'start': 5919.275, 'weight': 7, 'content': [{'end': 5930.951, 'text': 'is that the three planes intersect in the following way that two of the planes intersect in a line and then the third plane cuts that line so that you find one point,', 'start': 5919.275, 'duration': 11.676}, {'end': 5934.695, 'text': 'a single unique point, a common point for the three planes to intersect.', 'start': 5930.951, 'duration': 3.744}, {'end': 5940.146, 'text': 'And then the last thing that can happen and again it probably strikes a person as unlikely,', 'start': 5936.462, 'duration': 3.684}], 'summary': 'Three planes intersect to form a single unique point.', 'duration': 20.871, 'max_score': 5919.275, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI5919275.jpg'}, {'end': 6006.636, 'src': 'embed', 'start': 5977.389, 'weight': 4, 'content': [{'end': 5981.21, 'text': 'And then the third case that the three planes intersect in a single point.', 'start': 5977.389, 'duration': 3.821}, {'end': 5984.252, 'text': 'two of the planes intersect at a line and then the third plane cuts the line.', 'start': 5981.21, 'duration': 3.042}, {'end': 5991.754, 'text': "That's kind of, you say yourself, that's kind of the, the, you know, the case that I kind of expect to happen most of the time.", 'start': 5985.232, 'duration': 6.522}, {'end': 5992.415, 'text': "Yeah, that's right.", 'start': 5991.794, 'duration': 0.621}, {'end': 5998.097, 'text': "We use the word non-singular before, but anyway, that's kind of the kind of the random case that you think would happen most often.", 'start': 5992.475, 'duration': 5.622}, {'end': 5999.898, 'text': 'Okay All right.', 'start': 5999.177, 'duration': 0.721}, {'end': 6006.636, 'text': "So let me, uh, Let me go to today's material on length and angle measure.", 'start': 5999.918, 'duration': 6.718}], 'summary': 'Three planes intersecting in different cases, with one case being the most expected.', 'duration': 29.247, 'max_score': 5977.389, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI5977389.jpg'}], 'start': 4737.763, 'title': 'Linear systems and vectors', 'summary': 'Delves into the analysis of linear systems, identifying solutions for different echelon forms, as well as introducing vectors, their operations, and geometric interpretations. it also covers the vector expressions of lines and planes, and explores the geometric interpretation of linear systems in 3-space.', 'chapters': [{'end': 5022.324, 'start': 4737.763, 'title': 'Linear systems solutions analysis', 'summary': 'Discusses the analysis of linear systems, with focus on identifying the number of solutions for different echelon forms, including cases of no solution, infinitely many solutions, and unique solutions.', 'duration': 284.561, 'highlights': ['Identifying the number of solutions for different echelon forms', 'Analysis of contradictory equations', 'Creation of systems with specific solutions']}, {'end': 5453.432, 'start': 5023.885, 'title': 'Introduction to vectors and geometry', 'summary': 'Introduces the concept of vectors as objects consisting of magnitude and direction, and how they represent displacement in a geometric sense. it also covers the operations of addition and multiplication of vectors, highlighting their geometric interpretations and applications in higher dimensions.', 'duration': 429.547, 'highlights': ['Vectors as objects consisting of magnitude and direction represent displacement.', 'Operations of addition and multiplication of vectors are introduced, highlighting their geometric interpretations.', 'The concept of vectors is extended to higher dimensions, applicable to systems with hundreds of unknowns and equations.']}, {'end': 5735.208, 'start': 5453.823, 'title': 'Vector expression of a line and planes', 'summary': 'Discusses the vector expression of a line through the points (1, 2) and (3, 1), and the representation of a plane as the solution set of a linear system in a k-dimensional linear surface, emphasizing the concept of direction vector and linear algebra.', 'duration': 281.385, 'highlights': ['The vector expression of a line through the points (1, 2) and (3, 1) is represented as 1, 2 + t(2, -1), where t varies, illustrating the concept of direction vector for a line.', 'The representation of a plane as the solution set of a linear system is explained through the example of a generic plane consisting of endpoints of vectors in a set, highlighting the concept of k-dimensional linear surface.', 'The discussion emphasizes the linear nature of lines, planes, and k-dimensional linear surfaces, providing insight into the subject of linear algebra.']}, {'end': 5992.415, 'start': 5735.648, 'title': 'Geometric interpretation of linear systems', 'summary': 'Explores the geometric interpretation of linear systems in 3-space, demonstrating that one equation linear system forms a plane, while two or three planes can intersect in various configurations, including parallel, intersecting in a line, or intersecting in a single point.', 'duration': 256.767, 'highlights': ['The solution set for x + 2y + 3z = 4 is a one-equation linear system with two free variables, resulting in a solution set that looks like a particular vector plus y times a vector plus z times a vector.', 'The geometric interpretation of x + 2y + 3z = 4 and -x + 2y - z = 1 shows that they represent two-dimensional flat surfaces (planes) in 3-space.', "Two planes can either be parallel, intersect in a line, or have no common intersection, while it's not possible for two planes to intersect in exactly two points or one point.", 'Three planes can intersect in various ways, including being parallel, intersecting in a line, or intersecting in a single point, demonstrating the different configurations that can occur in 3-space.']}], 'duration': 1254.652, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI4737763.jpg', 'highlights': ['Creation of systems with specific solutions', 'Identifying the number of solutions for different echelon forms', 'Vectors as objects with magnitude and direction represent displacement', 'Operations of addition and multiplication of vectors introduced', 'The vector expression of a line through the points (1, 2) and (3, 1) is represented as 1, 2 + t(2, -1)', 'The solution set for x + 2y + 3z = 4 is a one-equation linear system with two free variables', 'The geometric interpretation of x + 2y + 3z = 4 and -x + 2y - z = 1 shows that they represent two-dimensional flat surfaces (planes) in 3-space', 'Two planes can either be parallel, intersect in a line, or have no common intersection', 'Three planes can intersect in various ways, including being parallel, intersecting in a line, or intersecting in a single point']}, {'end': 7673.211, 'segs': [{'end': 6553.106, 'src': 'embed', 'start': 6504.466, 'weight': 0, 'content': [{'end': 6509.75, 'text': 'and that can only happen in the case that one vector is a multiple of the other.', 'start': 6504.466, 'duration': 5.284}, {'end': 6518.043, 'text': 'For example, if v is length V over length U multiplied by U.', 'start': 6509.811, 'duration': 8.232}, {'end': 6522.164, 'text': 'And I mentioned a number of times here is that I was keen on this inequality here.', 'start': 6518.043, 'duration': 4.121}, {'end': 6526.945, 'text': "So that's why I did that particular proof.", 'start': 6522.624, 'duration': 4.321}, {'end': 6531.827, 'text': 'And it came up in the course of the triangle inequality.', 'start': 6527.606, 'duration': 4.221}, {'end': 6533.907, 'text': 'So I was interested in it for that purposes.', 'start': 6531.867, 'duration': 2.04}, {'end': 6537.528, 'text': 'That was a reason to go through that particular proof of the triangle inequality.', 'start': 6534.167, 'duration': 3.361}, {'end': 6541.238, 'text': 'Okay, so I said I was going to do length and angles.', 'start': 6539.477, 'duration': 1.761}, {'end': 6545.361, 'text': "We're going to need angles at some point, so that fits with length, so we're throwing it in here.", 'start': 6541.438, 'duration': 3.923}, {'end': 6548.703, 'text': 'So the angle between two vectors is given by that equation.', 'start': 6545.881, 'duration': 2.822}, {'end': 6549.824, 'text': 'Oops, my mouse is in the way.', 'start': 6548.743, 'duration': 1.081}, {'end': 6553.106, 'text': 'The angle between two vectors is given by that equation.', 'start': 6550.444, 'duration': 2.662}], 'summary': 'Vectors: relationship, triangle inequality, angles, equational representation.', 'duration': 48.64, 'max_score': 6504.466, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI6504466.jpg'}, {'end': 7435.683, 'src': 'embed', 'start': 7403.915, 'weight': 2, 'content': [{'end': 7405.597, 'text': 'Instead of just doing back substitution,', 'start': 7403.915, 'duration': 1.682}, {'end': 7412.885, 'text': "I'm going to continue doing row operations and I'm going to go on to get the system in some sense completely reduced.", 'start': 7405.597, 'duration': 7.288}, {'end': 7417.371, 'text': "Instead of having this junk up here, I'll turn those into zeros also.", 'start': 7413.286, 'duration': 4.085}, {'end': 7419.569, 'text': "So we'll start off.", 'start': 7418.649, 'duration': 0.92}, {'end': 7421.73, 'text': 'I have a particular process in mind.', 'start': 7419.609, 'duration': 2.121}, {'end': 7426.812, 'text': "I'll start off by here I have a 1 in front of x and a minus 3 in front of y and a 1 in front of z.", 'start': 7421.77, 'duration': 5.042}, {'end': 7430.053, 'text': 'I want to turn all the coefficients of the leading variables into ones.', 'start': 7426.812, 'duration': 3.241}, {'end': 7435.683, 'text': "So, of course, I'm going to take minus 1 third row 2.", 'start': 7430.553, 'duration': 5.13}], 'summary': 'Performing row operations to reduce coefficients, e.g., turning -3 into 1.', 'duration': 31.768, 'max_score': 7403.915, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI7403915.jpg'}, {'end': 7473.306, 'src': 'embed', 'start': 7447.516, 'weight': 6, 'content': [{'end': 7453.062, 'text': 'Before, I first used the X to eliminate, and then I used the Y to eliminate both of them down.', 'start': 7447.516, 'duration': 5.546}, {'end': 7456.898, 'text': "Here, I'm going to eliminate up.", 'start': 7454.517, 'duration': 2.381}, {'end': 7459.239, 'text': "When you eliminate up, you don't start with x.", 'start': 7457.358, 'duration': 1.881}, {'end': 7462.061, 'text': 'Instead, when you eliminate x, you start and you work right to left.', 'start': 7459.239, 'duration': 2.822}, {'end': 7464.041, 'text': "So I'm going to start with the z.", 'start': 7462.401, 'duration': 1.64}, {'end': 7470.485, 'text': "the z that's right here, and I'm going to use this row, row 3, to eliminate the minus 2 thirds z.", 'start': 7464.041, 'duration': 6.444}, {'end': 7473.306, 'text': "and I'm going to use this row row 3, to eliminate the minus z.", 'start': 7470.485, 'duration': 2.821}], 'summary': 'Using row operations to eliminate variables in a linear system.', 'duration': 25.79, 'max_score': 7447.516, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI7447516.jpg'}, {'end': 7548.643, 'src': 'embed', 'start': 7518.159, 'weight': 5, 'content': [{'end': 7519.32, 'text': "You've got to write them in some order.", 'start': 7518.159, 'duration': 1.161}, {'end': 7520.941, 'text': 'You might as well write them in the same order every time.', 'start': 7519.34, 'duration': 1.601}, {'end': 7527.533, 'text': "So when you do this, you've used z to eliminate both of the things above it.", 'start': 7522.731, 'duration': 4.802}, {'end': 7531.035, 'text': "So we're not only clearing out downstairs, we're clearing out upstairs also.", 'start': 7527.613, 'duration': 3.422}, {'end': 7534.016, 'text': "And a person can guess what's going to happen next.", 'start': 7532.195, 'duration': 1.821}, {'end': 7535.597, 'text': "I'm going to move from right to left.", 'start': 7534.036, 'duration': 1.561}, {'end': 7540.86, 'text': "I'm going to use this y to get rid of that y with a minus row 2, add to row 1.", 'start': 7535.857, 'duration': 5.003}, {'end': 7542.48, 'text': 'So there we go.', 'start': 7540.86, 'duration': 1.62}, {'end': 7548.643, 'text': "Now, this method has a disadvantage that there's more arithmetic than with Gauss's method.", 'start': 7543.941, 'duration': 4.702}], 'summary': "Using z eliminates both things above it, clears out upstairs and downstairs, more arithmetic than gauss's method.", 'duration': 30.484, 'max_score': 7518.159, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI7518159.jpg'}, {'end': 7636.446, 'src': 'embed', 'start': 7606.963, 'weight': 1, 'content': [{'end': 7610.245, 'text': 'Looks perfectly innocuous linear system, and it is perfectly innocuous.', 'start': 7606.963, 'duration': 3.282}, {'end': 7613.007, 'text': "So, of course, we recognize the Gauss's method.", 'start': 7610.625, 'duration': 2.382}, {'end': 7621.291, 'text': "It so happens that it takes only one row operation to do Gauss's method, and to get so that below this diagonal is all zeros.", 'start': 7613.367, 'duration': 7.924}, {'end': 7626.975, 'text': "We took one operation, and you get some other arithmetic, and it's not very interesting.", 'start': 7623.173, 'duration': 3.802}, {'end': 7633.659, 'text': 'Next step, so after you clear out below the diagonal, next step is to turn the diagonal into ones.', 'start': 7627.695, 'duration': 5.964}, {'end': 7636.446, 'text': 'So I have to turn the two into a one.', 'start': 7634.645, 'duration': 1.801}], 'summary': "Gauss's method takes one row operation to clear out below the diagonal, and then turns the diagonal into ones.", 'duration': 29.483, 'max_score': 7606.963, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI7606963.jpg'}], 'start': 5992.475, 'title': 'Vector operations and linear systems', 'summary': 'Covers vector length calculation using the pythagorean theorem, normalizing vector length to 1, dot product properties, triangle inequality, angle calculation, linear system solution set in r3, and gauss-jordan reduction method for solving linear systems, emphasizing concepts like infinitely many solutions and advantages of reduced form.', 'chapters': [{'end': 6314.856, 'start': 5992.475, 'title': 'Vector length and dot product', 'summary': 'Covers the calculation of vector length using the pythagorean theorem, the process of normalizing vector length to 1, and the definition and properties of the dot product, including its calculation and the triangle inequality, which states that the length of the sum of two vectors is less than or equal to the sum of their individual lengths.', 'duration': 322.381, 'highlights': ['The process of normalizing vector length to 1 involves dividing the vector by its size, which results in a new vector with a length of 1.', 'The dot product of two n-component real vectors is calculated by multiplying their corresponding components and then summing up the results, resulting in a scalar quantity.', 'The chapter also covers the calculation of vector length using the Pythagorean theorem, which involves squaring each component, summing them up, and taking the square root.']}, {'end': 6719.741, 'start': 6314.876, 'title': 'Triangle inequality and angle calculation', 'summary': 'Discusses the triangle inequality and angle calculation, illustrating the relationship between vectors through examples and proofs, emphasizing the inequality u dot v <= length u * length v and the calculation of the angle between two vectors using the equation u dot v / (length u * length v).', 'duration': 404.865, 'highlights': ['The inequality u dot v <= length u * length v is emphasized through examples and proofs, demonstrating its significance in vector calculations.', 'The calculation of the angle between two vectors is explained using the equation u dot v / (length u * length v), showcasing the connection between vectors and trigonometry.', 'The concept of parallel and perpendicular vectors is linked to the dot product, where perpendicular vectors have a dot product of zero, and parallel vectors have a specific relationship based on the angle calculation equation.']}, {'end': 7010.196, 'start': 6719.741, 'title': 'Linear system solution set in r3', 'summary': 'Discusses the sketch of a solution set of a linear system on a plane in r3, emphasizing the distinction between the homogeneous and particular parts and the concept of infinitely many solutions.', 'duration': 290.455, 'highlights': ['The solution set is a plane in R3, represented as a set of all vectors x, y, z, which can be written in the form 1, 1, 2 plus vector times a letter and another vector times a letter.', 'The distinction between the homogeneous part and the particular part is emphasized, where the homogeneous parts lie entirely within the plane, while the particular reaches out from the floor and just touches the plane.', 'The discussion focuses on the concept of infinitely many solutions, attributed to the homogeneous part, leading to the understanding that the interesting aspect of the solution set is the existence of infinitely many solutions.']}, {'end': 7673.211, 'start': 7011.874, 'title': 'Gauss-jordan reduction and solution sets', 'summary': 'Discusses the application of gauss-jordan reduction method to solve linear systems, emphasizing the process of turning coefficients into ones and eliminating upwards, leading to a reduced form with advantages and more arithmetic but a clear solution.', 'duration': 661.337, 'highlights': ['The process of turning coefficients of leading variables into ones is emphasized as a key step in the Gauss-Jordan reduction method.', 'The technique of eliminating upwards, starting from the bottom and working upwards, is introduced as a new component of the Gauss-Jordan reduction.', 'The method of pivoting on an entry is mentioned as a technique to clear out entire columns, leading to a clearer final answer.']}], 'duration': 1680.736, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI5992475.jpg', 'highlights': ['The process of normalizing vector length to 1 involves dividing the vector by its size, resulting in a new vector with a length of 1.', 'The dot product of two n-component real vectors is calculated by multiplying their corresponding components and then summing up the results, resulting in a scalar quantity.', 'The calculation of vector length using the Pythagorean theorem involves squaring each component, summing them up, and taking the square root.', 'The inequality u dot v <= length u * length v is emphasized through examples and proofs, demonstrating its significance in vector calculations.', 'The calculation of the angle between two vectors is explained using the equation u dot v / (length u * length v), showcasing the connection between vectors and trigonometry.', 'The solution set is a plane in R3, represented as a set of all vectors x, y, z, which can be written in the form 1, 1, 2 plus vector times a letter and another vector times a letter.', 'The process of turning coefficients of leading variables into ones is emphasized as a key step in the Gauss-Jordan reduction method.', 'The concept of infinitely many solutions is attributed to the homogeneous part, leading to the understanding that the interesting aspect of the solution set is the existence of infinitely many solutions.']}, {'end': 8698.438, 'segs': [{'end': 7701.936, 'src': 'embed', 'start': 7673.251, 'weight': 8, 'content': [{'end': 7676.53, 'text': 'Takes minus 3 has row 3 plus row 2.', 'start': 7673.251, 'duration': 3.279}, {'end': 7677.03, 'text': 'There we go.', 'start': 7676.53, 'duration': 0.5}, {'end': 7682.031, 'text': 'And then to use the one to eliminate above it takes row two plus row one.', 'start': 7677.97, 'duration': 4.061}, {'end': 7684.512, 'text': 'And I end up with a system.', 'start': 7682.071, 'duration': 2.441}, {'end': 7685.532, 'text': 'Something was in my way.', 'start': 7684.732, 'duration': 0.8}, {'end': 7685.892, 'text': "I'm sorry.", 'start': 7685.592, 'duration': 0.3}, {'end': 7694.014, 'text': 'So I end up with a system that has this property that below the diagonal is all zeros and above the diagonal is all zeros.', 'start': 7686.312, 'duration': 7.702}, {'end': 7698.775, 'text': "And there's some other junk simply because there are more unknowns than there are equations.", 'start': 7694.094, 'duration': 4.681}, {'end': 7701.936, 'text': "So you can't have the case that every unknown leads an equation.", 'start': 7698.815, 'duration': 3.121}], 'summary': 'Row operations used to create a system with zeros below and above diagonal.', 'duration': 28.685, 'max_score': 7673.251, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI7673251.jpg'}, {'end': 7940.898, 'src': 'embed', 'start': 7895.229, 'weight': 3, 'content': [{'end': 7897.51, 'text': 'Elementary row operations are reversible.', 'start': 7895.229, 'duration': 2.281}, {'end': 7902.726, 'text': "When I do Gauss' method here, or Gauss-Jordan method, ah, there we go.", 'start': 7898.544, 'duration': 4.182}, {'end': 7911.17, 'text': "When I do Gauss' method or Gauss-Jordan method, I have a mental image that I start with some kind of mess and I end with something that's very neat.", 'start': 7902.986, 'duration': 8.184}, {'end': 7915.618, 'text': 'So that is to say, I have a mental image that I proceed,', 'start': 7912.216, 'duration': 3.402}, {'end': 7922.581, 'text': 'this one comes after and then comes after and then comes after and then finally I arrive at my goal.', 'start': 7915.618, 'duration': 6.963}, {'end': 7929.524, 'text': 'So I have a sense of some kind of, that they proceed in some order.', 'start': 7922.721, 'duration': 6.803}, {'end': 7933.986, 'text': 'But in fact that proves to not really be perfectly right.', 'start': 7930.104, 'duration': 3.882}, {'end': 7940.898, 'text': "If I do Gauss's method on a matrix, any of the Gauss's method operations.", 'start': 7936.135, 'duration': 4.763}], 'summary': "Elementary row operations in gauss' method lead to orderly transformation of matrices.", 'duration': 45.669, 'max_score': 7895.229, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI7895229.jpg'}, {'end': 8090.502, 'src': 'embed', 'start': 8061.687, 'weight': 2, 'content': [{'end': 8071.578, 'text': "I'm asking you to not necessarily think of the reduced matrices as somehow later or lesser.", 'start': 8061.687, 'duration': 9.891}, {'end': 8073.48, 'text': "That in fact, they're the same.", 'start': 8072.319, 'duration': 1.161}, {'end': 8076.143, 'text': 'They contain exactly the same information as the original matrix.', 'start': 8073.5, 'duration': 2.643}, {'end': 8082.639, 'text': 'So the way that you write this down formally is.', 'start': 8080.078, 'duration': 2.561}, {'end': 8090.502, 'text': 'you describe the phrase that the between matrices reduces to via elementary rule operations is an equivalence relation.', 'start': 8082.639, 'duration': 7.863}], 'summary': 'Reduced matrices contain same information as original matrix.', 'duration': 28.815, 'max_score': 8061.687, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI8061687.jpg'}, {'end': 8158.648, 'src': 'embed', 'start': 8129.636, 'weight': 5, 'content': [{'end': 8135.398, 'text': "We can in fact go from this matrix over to this matrix, and I do that sometimes when I'm making homework problems.", 'start': 8129.636, 'duration': 5.762}, {'end': 8143.861, 'text': 'That in fact all of these matrices 1, 2, 3, 4, 5 are inter-reducible.', 'start': 8137.279, 'duration': 6.582}, {'end': 8145.942, 'text': "They're all collected together in some sense.", 'start': 8143.881, 'duration': 2.061}, {'end': 8148.143, 'text': "They're all somehow the same.", 'start': 8146.242, 'duration': 1.901}, {'end': 8150.844, 'text': 'They contain the same information as each other.', 'start': 8148.203, 'duration': 2.641}, {'end': 8156.427, 'text': 'So they are equivalent in some sense.', 'start': 8154.045, 'duration': 2.382}, {'end': 8158.648, 'text': 'The technical jargon is equivalence relation.', 'start': 8156.787, 'duration': 1.861}], 'summary': 'Matrices 1, 2, 3, 4, 5 are inter-reducible and equivalent, forming an equivalence relation.', 'duration': 29.012, 'max_score': 8129.636, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI8129636.jpg'}, {'end': 8211.023, 'src': 'embed', 'start': 8185.246, 'weight': 1, 'content': [{'end': 8191.069, 'text': "I've broken the collection of all matrices up into different equivalence classes.", 'start': 8185.246, 'duration': 5.823}, {'end': 8196.71, 'text': 'Inside of a particular class are matrices that are equivalent to each other.', 'start': 8191.689, 'duration': 5.021}, {'end': 8203.915, 'text': 'So on the prior slide, I had five matrices, one, two, three, four, five, All of them go in the same class.', 'start': 8197.092, 'duration': 6.823}, {'end': 8206.959, 'text': 'They all can be interrelated to each the other.', 'start': 8203.956, 'duration': 3.003}, {'end': 8211.023, 'text': "Isn't that a terrible phrase? They're all interrelated.", 'start': 8208.281, 'duration': 2.742}], 'summary': 'Matrices are grouped into equivalence classes, with five matrices in one class, all interrelated.', 'duration': 25.777, 'max_score': 8185.246, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI8185246.jpg'}, {'end': 8366.954, 'src': 'embed', 'start': 8320.111, 'weight': 0, 'content': [{'end': 8322.293, 'text': 'And of course, I can give all kinds of answers.', 'start': 8320.111, 'duration': 2.182}, {'end': 8325.895, 'text': "I can give a matrix with itself, for example, that'd be silly, but technically correct.", 'start': 8322.373, 'duration': 3.522}, {'end': 8332.217, 'text': 'But I want to get across something that you know, that makes the lesson for later on in the course.', 'start': 8326.275, 'duration': 5.942}, {'end': 8340.599, 'text': 'So the idea here is I want to look at two matrices that are row equivalent to contain some concept.', 'start': 8332.677, 'duration': 7.922}, {'end': 8346.081, 'text': 'So the concept is that matrices are row equivalent when they contain somehow the same information.', 'start': 8340.759, 'duration': 5.322}, {'end': 8350.342, 'text': 'So my example of two matrices that are row equivalent is going to be this.', 'start': 8347.001, 'duration': 3.341}, {'end': 8354.102, 'text': 'How about 1, 1, 3, and 0, 1, 1 from my first matrix.', 'start': 8350.642, 'duration': 3.46}, {'end': 8366.954, 'text': "And you know what? I'm going to make it an augmented matrix to make it look like a linear system, to make it look like I'm solving a linear system.", 'start': 8359.85, 'duration': 7.104}], 'summary': 'Matrices are row equivalent when they contain the same information, as demonstrated by example 1, 1, 3, and 0, 1, 1.', 'duration': 46.843, 'max_score': 8320.111, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI8320111.jpg'}], 'start': 7673.251, 'title': 'Gauss-jordan method and reduction', 'summary': 'Discusses the gauss-jordan method for solving linear equations, resulting in a matrix with zeros above and below the diagonal, and highlights benefits such as lower step count, reversible operations, equivalence of reduced matrices, and the importance of row equivalence in capturing linear content of rows.', 'chapters': [{'end': 7835.36, 'start': 7673.251, 'title': 'Gauss-jordan method', 'summary': 'Discusses the gauss-jordan method for solving a system of linear equations, which results in a matrix with zeros below and above the diagonal, and requires more arithmetic but facilitates a straightforward transcription to obtain the solution set.', 'duration': 162.109, 'highlights': ['Gauss-Jordan method results in a matrix with zeros below and above the diagonal', 'More arithmetic is required for the Gauss-Jordan method', 'Facilitates a straightforward transcription to obtain the solution set']}, {'end': 8268.01, 'start': 7837.834, 'title': 'Gauss-jordan reduction benefits', 'summary': 'Discusses the benefits of gauss-jordan reduction, highlighting its lower step count and reversible elementary row operations, leading to the equivalence of reduced matrices and the ability to categorize matrices into row equivalent classes.', 'duration': 430.176, 'highlights': ["Gauss' method takes fewer steps than Gauss-Jordan reduction, despite the additional arithmetic involved.", "Elementary row operations in Gauss' method are reversible, allowing the original matrix to be restored after operations such as row swaps or multiplication by a non-zero constant.", 'Matrices reduced to each other are equivalent with respect to row operations, forming row equivalent classes.']}, {'end': 8698.438, 'start': 8268.87, 'title': 'Gauss-jordan reduction and row equivalence', 'summary': 'Discusses the concept of row equivalence, providing examples of matrices that are row equivalent and not row equivalent, highlighting the importance of row equivalence in capturing the linear content of rows.', 'duration': 429.568, 'highlights': ['The concept of row equivalence is illustrated through the example of two matrices, 1, 1, 3, 0, 1, 1 and 1, 0, 2, 0, 1, 1, which are row equivalent and represent linear systems with the same solution, x equals 2, y equals 1.', 'A clear explanation is given on the non-row equivalence of the matrices 1, 0, 0, 0, 0, 0 and 0, 0, 0, 0, 1, 0, emphasizing the impossibility of transforming one into the other through row operations due to the absence of ones in the respective columns.', 'The comparison of matrices 1, 0, 0, 0, 0, 0 and 1, 3, 3, 0, 1, 1 highlights the difference in their linear content, demonstrating that the former is not row equivalent to the latter due to the inability to transform one into the other through row operations.', 'The importance of row equivalence in capturing the linear content of rows and determining whether two matrices contain the same information is emphasized, laying the groundwork for the upcoming precise definitions.']}], 'duration': 1025.187, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI7673251.jpg', 'highlights': ['Gauss-Jordan method results in a matrix with zeros below and above the diagonal', 'Facilitates a straightforward transcription to obtain the solution set', "Elementary row operations in Gauss' method are reversible, allowing the original matrix to be restored after operations", 'Matrices reduced to each other are equivalent with respect to row operations, forming row equivalent classes', 'The concept of row equivalence is illustrated through the example of two matrices, 1, 1, 3, 0, 1, 1 and 1, 0, 2, 0, 1, 1, which are row equivalent and represent linear systems with the same solution, x equals 2, y equals 1', 'The importance of row equivalence in capturing the linear content of rows and determining whether two matrices contain the same information is emphasized, laying the groundwork for the upcoming precise definitions', "Gauss' method takes fewer steps than Gauss-Jordan reduction, despite the additional arithmetic involved", 'More arithmetic is required for the Gauss-Jordan method', 'The comparison of matrices 1, 0, 0, 0, 0, 0 and 1, 3, 3, 0, 1, 1 highlights the difference in their linear content, demonstrating that the former is not row equivalent to the latter due to the inability to transform one into the other through row operations', 'A clear explanation is given on the non-row equivalence of the matrices 1, 0, 0, 0, 0, 0 and 0, 0, 0, 0, 1, 0, emphasizing the impossibility of transforming one into the other through row operations due to the absence of ones in the respective columns']}, {'end': 11703.59, 'segs': [{'end': 9072.564, 'src': 'embed', 'start': 9037.685, 'weight': 2, 'content': [{'end': 9042.369, 'text': 'And I take D1 times the first all the way down to DM times the last.', 'start': 9037.685, 'duration': 4.684}, {'end': 9047.353, 'text': 'And then just distribute D1 over the Cs through DM over the Cs.', 'start': 9043.05, 'duration': 4.303}, {'end': 9052.518, 'text': 'Gather together all the X1s, gather together all the X1s and the XNs.', 'start': 9047.734, 'duration': 4.784}, {'end': 9055.28, 'text': 'And I end up with a linear combination of the Xs.', 'start': 9053.159, 'duration': 2.121}, {'end': 9062.306, 'text': "So this proof right here is simply putting into I's and J's what this example has.", 'start': 9055.621, 'duration': 6.685}, {'end': 9072.564, 'text': "in numbers, so that the proof doesn't contain anything, any content that the example doesn't contain, except that it shows how you do the proof.", 'start': 9063.095, 'duration': 9.469}], 'summary': 'Distribute d1 to dm over cs to get linear combination of xs.', 'duration': 34.879, 'max_score': 9037.685, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI9037685.jpg'}, {'end': 9222.166, 'src': 'embed', 'start': 9194.013, 'weight': 6, 'content': [{'end': 9196.754, 'text': 'All right, so I wrote down a matrix that is in echelon form.', 'start': 9194.013, 'duration': 2.741}, {'end': 9202.717, 'text': 'I want to show that no row is a linear combination of the others.', 'start': 9198.855, 'duration': 3.862}, {'end': 9212.041, 'text': 'It so happens that, in order to best illustrate the proof, the thing I should do is to show that the middle row 0, 5, 6, 7, 0, 5, 6, 7,', 'start': 9203.437, 'duration': 8.604}, {'end': 9215.843, 'text': 'is not a linear combination of 1, 2, 3, 4, and 0, 0, 0, 8..', 'start': 9212.041, 'duration': 3.802}, {'end': 9222.166, 'text': "Okay, so I want to show that there's no linear combination that gives you this.", 'start': 9215.843, 'duration': 6.323}], 'summary': 'Matrix in echelon form; proving row independence with specific example.', 'duration': 28.153, 'max_score': 9194.013, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI9194013.jpg'}, {'end': 10304.061, 'src': 'embed', 'start': 10275.964, 'weight': 0, 'content': [{'end': 10279.566, 'text': '2, 3, 4, and 5 describe something about linear combinations.', 'start': 10275.964, 'duration': 3.602}, {'end': 10289.771, 'text': "So for example, we will consider that if you had some kind of plus operation that was not commutative, we won't think of that as a linear combination.", 'start': 10280.446, 'duration': 9.325}, {'end': 10292.773, 'text': 'That linear combination, really the plus operation has to be commutative.', 'start': 10289.811, 'duration': 2.962}, {'end': 10295.275, 'text': "If you add two vectors, that's commutative.", 'start': 10293.314, 'duration': 1.961}, {'end': 10297.377, 'text': "If you add two matrices, that's commutative.", 'start': 10295.435, 'duration': 1.942}, {'end': 10304.061, 'text': "So we're thinking of the plus operation has to be commutative or else we won't consider it a legal case.", 'start': 10297.597, 'duration': 6.464}], 'summary': 'Linear combinations involve commutative plus operations with vectors and matrices.', 'duration': 28.097, 'max_score': 10275.964, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI10275964.jpg'}, {'end': 11199.907, 'src': 'embed', 'start': 11168.531, 'weight': 1, 'content': [{'end': 11170.893, 'text': 'And the answer is it says there exists.', 'start': 11168.531, 'duration': 2.362}, {'end': 11172.854, 'text': 'So we produced it.', 'start': 11171.853, 'duration': 1.001}, {'end': 11179.579, 'text': "We produced the thing, which, when you do the defined operation, doesn't change the vector.", 'start': 11172.974, 'duration': 6.605}, {'end': 11192.185, 'text': 'For condition 2, we had to write down the left-hand side v plus w and the right-hand side w plus v and check that the two of them are the same vector.', 'start': 11181.462, 'duration': 10.723}, {'end': 11199.907, 'text': "So for many of these cases here, for example, 9, 8, for many of these cases, it's extremely easy to check.", 'start': 11193.165, 'duration': 6.742}], 'summary': 'Produced a non-changing vector and verified equality for multiple cases.', 'duration': 31.376, 'max_score': 11168.531, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI11168531.jpg'}, {'end': 11685.41, 'src': 'embed', 'start': 11630.193, 'weight': 3, 'content': [{'end': 11632.734, 'text': "Okay, so that's the idea of a vector space.", 'start': 11630.193, 'duration': 2.541}, {'end': 11634.555, 'text': "That's the concept behind a vector space.", 'start': 11632.774, 'duration': 1.781}, {'end': 11639.776, 'text': "So let's see here.", 'start': 11635.155, 'duration': 4.621}, {'end': 11650.559, 'text': "So I want to talk today about some more examples of vector spaces and try to get across again some more of the concepts that illustrate here what it's about.", 'start': 11639.856, 'duration': 10.703}, {'end': 11661.657, 'text': "So the main concept, there's a lot of detail here, but the main concept in this detail is line 1 and line 6.", 'start': 11653.248, 'duration': 8.409}, {'end': 11668.545, 'text': 'A vector space is a place where linear combinations take place, and these other conditions the conditions that are not 1 and 6,', 'start': 11661.657, 'duration': 6.888}, {'end': 11672.71, 'text': 'really serve to define what I mean by linear combination.', 'start': 11668.545, 'duration': 4.165}, {'end': 11675.599, 'text': 'you can multiply by a scalar.', 'start': 11673.817, 'duration': 1.782}, {'end': 11684.489, 'text': 'When you multiply by a scalar, for example, the multiplication distributes over addition, or the addition of two things is the same.', 'start': 11675.679, 'duration': 8.81}, {'end': 11685.41, 'text': "either way, it's commutative.", 'start': 11684.489, 'duration': 0.921}], 'summary': 'Vector spaces are places for linear combinations with conditions like distributive and commutative properties.', 'duration': 55.217, 'max_score': 11630.193, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI11630193.jpg'}], 'start': 8698.458, 'title': 'Linear algebra fundamentals', 'summary': "Covers topics like linear combination lemma, gauss's method, proving non-linearity in echelon form matrices, reduced echelon form, vector spaces, and their properties, providing examples and applications, showcasing the importance of these concepts in linear algebra.", 'chapters': [{'end': 9192.263, 'start': 8698.458, 'title': "Linear combination lemma & gauss's method", 'summary': "Introduces the importance of the linear combination lemma in illustrating gauss's method, showing its application in transforming matrices, and highlighting the concept that a linear combination of linear combinations is a linear combination.", 'duration': 493.805, 'highlights': ["The linear combination lemma is crucial for understanding Gauss's method, as it demonstrates the transformation of matrices through operations such as eliminating below the diagonal, changing the diagonal to ones, changing leading entries to ones, and eliminating above, emphasizing the significance of paying special attention to this concept.", "Gauss's method acts by taking linear combinations of the rows of a matrix, with the resulting matrix containing linear combinations of the rows of the initial matrix, illustrating the fundamental principle that a linear combination of linear combinations is a linear combination.", 'The concept that a linear combination of linear combinations is a linear combination is exemplified through the demonstration of combining two linear combinations and obtaining a linear combination, emphasizing the importance of this small observation in understanding linear algebra and its applications.', 'The chapter also emphasizes the significance of echelon form matrices, highlighting that in echelon form, none of the non-zero rows is a linear combination of the others, with the importance of understanding this concept for further exploration in the course.']}, {'end': 9672.198, 'start': 9194.013, 'title': 'Proving non-linearity in an echelon form matrix', 'summary': 'Demonstrates the process of proving that no row in an echelon form matrix is a linear combination of the others using a detailed example, highlighting the importance of gaussian elimination and row equivalence in determining linear relationships between matrices.', 'duration': 478.185, 'highlights': ["Gauss's method acts by taking linear combinations of the rows to systematically eliminate any linear relationships among the rows, providing a deeper understanding of the linear content of a matrix.", 'The example illustrates the process of proving non-linearity by showing the steps involved in identifying and eliminating linear relationships through Gaussian elimination and comparing reduced echelon forms, emphasizing the importance of unique reduced echelon form matrices in determining row equivalence.', 'The proof involves mathematical induction and rewards close study, contributing to a deeper understanding of linear algebra and its applications.']}, {'end': 10185.083, 'start': 9672.538, 'title': 'Reduced echelon form and linear combinations', 'summary': 'Introduces the concept of reduced echelon form matrices, which serve as canonical representatives for classes of matrices, and emphasizes the use of gauss-jordan elimination to determine row equivalence. additionally, it delves into the expansion of linear combinations to include vectors and matrices, showcasing the application of gaussian elimination in solving linear combination equations.', 'duration': 512.545, 'highlights': ['Gauss-Jordan elimination is used to determine when matrices are row-equivalent, with even minor differences indicating inequivalence.', 'Introduction of the concept of reduced echelon form matrices as canonical representatives for classes of matrices, serving as a way to recognize and characterize different classes of matrices.', 'Expansion of linear combinations to include vectors and matrices, demonstrating the application of Gaussian elimination in solving linear combination equations.']}, {'end': 10992.068, 'start': 10185.123, 'title': 'Understanding vector spaces', 'summary': 'Explains the properties and conditions of a vector space, with a focus on closure and scalar multiplication, and provides an example of a line in a plane that satisfies all the conditions for being a vector space.', 'duration': 806.945, 'highlights': ['The chapter explains the properties and conditions of a vector space, with a focus on closure and scalar multiplication.', 'Provides an example of a line in a plane that satisfies all the conditions for being a vector space.', 'Explains the conditions of closure and scalar multiplication in detail, providing step-by-step explanations and examples for each condition.']}, {'end': 11703.59, 'start': 10993.229, 'title': 'Vector space definition and examples', 'summary': 'Covers the 10 conditions to check for a vector space, emphasizing the importance of conditions 1 and 6, which ensure closure under vector addition and scalar multiplication. it also provides examples of linear combinations in a vector space and contrasts it with a non-vector space, highlighting the key conceptual idea of a vector space being characterized by closure under linear combinations.', 'duration': 710.361, 'highlights': ['The chapter emphasizes the importance of conditions 1 and 6, which ensure closure under vector addition and scalar multiplication, as they are the crux of the matter (relevance: 5)', 'Examples of linear combinations in a vector space and a non-vector space are provided to illustrate closure under linear combinations (relevance: 4)', 'The chapter covers the 10 conditions to check for a vector space, with conditions 1 and 6 being the most important ones (relevance: 3)', 'The definition of a vector space is discussed, with a focus on the conceptual idea of closure under linear combinations (relevance: 2)', 'The chapter provides examples of different kinds of vector spaces, such as 2x2 matrices and 3D vectors, to illustrate the diversity of vector spaces (relevance: 1)']}], 'duration': 3005.132, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI8698458.jpg', 'highlights': ["The linear combination lemma is crucial for understanding Gauss's method, as it demonstrates the transformation of matrices through operations such as eliminating below the diagonal, changing the diagonal to ones, changing leading entries to ones, and eliminating above, emphasizing the significance of paying special attention to this concept.", "Gauss's method acts by taking linear combinations of the rows of a matrix, with the resulting matrix containing linear combinations of the rows of the initial matrix, illustrating the fundamental principle that a linear combination of linear combinations is a linear combination.", 'The example illustrates the process of proving non-linearity by showing the steps involved in identifying and eliminating linear relationships through Gaussian elimination and comparing reduced echelon forms, emphasizing the importance of unique reduced echelon form matrices in determining row equivalence.', 'The chapter explains the properties and conditions of a vector space, with a focus on closure and scalar multiplication.', 'The chapter emphasizes the importance of conditions 1 and 6, which ensure closure under vector addition and scalar multiplication, as they are the crux of the matter (relevance: 5)', 'Gauss-Jordan elimination is used to determine when matrices are row-equivalent, with even minor differences indicating inequivalence.', 'Introduction of the concept of reduced echelon form matrices as canonical representatives for classes of matrices, serving as a way to recognize and characterize different classes of matrices.', 'The chapter provides examples of different kinds of vector spaces, such as 2x2 matrices and 3D vectors, to illustrate the diversity of vector spaces (relevance: 1)']}, {'end': 12784.386, 'segs': [{'end': 11749.963, 'src': 'embed', 'start': 11723.669, 'weight': 2, 'content': [{'end': 11731.453, 'text': 'And now I want to look at a new, instead of a line through the origin in R2, I want to look at a new collection of vectors.', 'start': 11723.669, 'duration': 7.784}, {'end': 11734.915, 'text': 'And this is going to be a plane through the origin in R3.', 'start': 11731.493, 'duration': 3.422}, {'end': 11740.078, 'text': 'The plane 2x plus y plus 3z equals 0 through the origin in R3.', 'start': 11735.875, 'duration': 4.203}, {'end': 11742.719, 'text': 'I have it drawn there in yellow.', 'start': 11740.098, 'duration': 2.621}, {'end': 11747.081, 'text': 'And I drew some vectors from the plane onto it.', 'start': 11744.22, 'duration': 2.861}, {'end': 11749.963, 'text': "I had the computer draw that because I'm not a very good drawer.", 'start': 11747.262, 'duration': 2.701}], 'summary': 'Introduction to a new collection of vectors in a plane through the origin in r3 with the equation 2x + y + 3z = 0.', 'duration': 26.294, 'max_score': 11723.669, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI11723669.jpg'}, {'end': 11818.651, 'src': 'embed', 'start': 11794.257, 'weight': 0, 'content': [{'end': 11800.261, 'text': 'Translating that into the geometry here, it says that two vectors in the plane add to make a vector in the plane.', 'start': 11794.257, 'duration': 6.004}, {'end': 11806.487, 'text': "I'm going to confirm that algebraically, but before I do the algebra, here's the geometry.", 'start': 11801.826, 'duration': 4.661}, {'end': 11808.448, 'text': "There's a red vector in the plane.", 'start': 11806.888, 'duration': 1.56}, {'end': 11810.008, 'text': "There's a red vector in the plane.", 'start': 11808.488, 'duration': 1.52}, {'end': 11817.031, 'text': 'Both of those sit, not just their tip in the plane, but their entire body sits in the plane.', 'start': 11811.169, 'duration': 5.862}, {'end': 11818.651, 'text': 'Remember, this is a plane through the origin.', 'start': 11817.071, 'duration': 1.58}], 'summary': 'Two vectors in a plane add to make a vector in the plane.', 'duration': 24.394, 'max_score': 11794.257, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI11794257.jpg'}, {'end': 12000.157, 'src': 'embed', 'start': 11972.606, 'weight': 1, 'content': [{'end': 11977.37, 'text': "and after some amount of head scratching here it's not very much head scratching you get zero on the other side.", 'start': 11972.606, 'duration': 4.764}, {'end': 11984.916, 'text': "All right, so of course, we're also going to verify the same thing for scalar multiplication.", 'start': 11977.39, 'duration': 7.526}, {'end': 11987.878, 'text': 'So I start off with a vector that is in the space.', 'start': 11985.516, 'duration': 2.362}, {'end': 11994.183, 'text': 'In this particular case, being in the space means two times its first, plus its second, plus three times its third, makes zero.', 'start': 11988.058, 'duration': 6.125}, {'end': 11997.355, 'text': 'and I multiply it by a scalar.', 'start': 11995.734, 'duration': 1.621}, {'end': 12000.157, 'text': "You can't multiply it by 7 and check just 7.", 'start': 11997.575, 'duration': 2.582}], 'summary': 'Verifying scalar multiplication in a vector space.', 'duration': 27.551, 'max_score': 11972.606, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI11972606.jpg'}, {'end': 12316.978, 'src': 'embed', 'start': 12291.053, 'weight': 3, 'content': [{'end': 12296.277, 'text': 'The concept here is that vector spaces are about taking linear combinations.', 'start': 12291.053, 'duration': 5.224}, {'end': 12301.021, 'text': 'It makes perfectly good sense to take a linear combination of these vectors.', 'start': 12296.737, 'duration': 4.284}, {'end': 12306.047, 'text': 'It makes perfectly good sense to take a linear combination of 1 plus 2x plus 3x squared.', 'start': 12301.903, 'duration': 4.144}, {'end': 12309.21, 'text': "You call that a vector in this context because it's a member of a vector space.", 'start': 12306.067, 'duration': 3.143}, {'end': 12316.978, 'text': 'It makes perfectly good sense to take 4 times that, a minus 1 fifth times 10 plus 5x squared.', 'start': 12309.791, 'duration': 7.187}], 'summary': 'Vector spaces involve taking linear combinations of vectors, such as 1 plus 2x plus 3x squared, and 4 times that, as members of the vector space.', 'duration': 25.925, 'max_score': 12291.053, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI12291053.jpg'}, {'end': 12583.586, 'src': 'embed', 'start': 12553.291, 'weight': 4, 'content': [{'end': 12556.212, 'text': 'You can have a vector space that has only one element.', 'start': 12553.291, 'duration': 2.921}, {'end': 12567.076, 'text': 'That can happen and I listed as an example the vector space that contains only the too-tall zero vector and the plus and times operations are pretty dopey there,', 'start': 12556.252, 'duration': 10.824}, {'end': 12569.817, 'text': "but it's still perfectly possible and satisfies all the rules.", 'start': 12567.076, 'duration': 2.741}, {'end': 12572.377, 'text': 'You call that the trivial space for the obvious reason,', 'start': 12570.137, 'duration': 2.24}, {'end': 12578.5, 'text': "and we'll have occasion sometimes to mention the trivial space for much the same way that you have occasion in high school algebra to mention the number zero.", 'start': 12572.377, 'duration': 6.123}, {'end': 12579.74, 'text': 'It just comes up sometimes.', 'start': 12578.6, 'duration': 1.14}, {'end': 12583.586, 'text': 'Next up is a small lemma.', 'start': 12581.866, 'duration': 1.72}], 'summary': 'A vector space can have only one element, known as the trivial space.', 'duration': 30.295, 'max_score': 12553.291, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI12553291.jpg'}], 'start': 11703.59, 'title': 'Vector spaces in r3', 'summary': 'Delves into the concept of different vector spaces, including planes in r3 with the equation 2x + y + 3z = 0, closure under addition and scalar multiplication, and linear combinations illustrated through examples of quadratic polynomials and 3x3 matrices.', 'chapters': [{'end': 11749.963, 'start': 11703.59, 'title': 'Vector spaces and planes in r3', 'summary': 'Discusses the concept of different kinds of vector spaces, highlighting the exploration of a plane through the origin in r3 represented by the equation 2x + y + 3z = 0, with illustrated vectors, and the comparison with a line in r2.', 'duration': 46.373, 'highlights': ['The chapter explores different kinds of vector spaces, emphasizing the introduction of a plane through the origin in R3 represented by the equation 2x + y + 3z = 0.', 'Illustrates the comparison between a line through the origin in R2 and the newly introduced plane in R3, providing visual representations of vectors on the plane.', "The speaker's approach involves using computer-generated illustrations to aid in the understanding of the concept, acknowledging limitations in drawing abilities."]}, {'end': 12290.973, 'start': 11752.009, 'title': 'Vector space and natural operations', 'summary': 'Discusses the conditions for closure under addition and scalar multiplication in vector spaces, illustrating how the sum and scalar multiple of vectors in a plane through the origin are also in the plane, and the vector spaces of n-tall vectors and quadratic polynomials.', 'duration': 538.964, 'highlights': ['The sum of vectors in a plane through the origin results in a vector in the same plane, verified algebraically and geometrically, ensuring closure under addition.', 'The scalar multiple of a vector in a plane through the origin remains in the same plane, confirmed by checking for an arbitrary scalar, satisfying the closure under scalar multiplication.', 'The collection of n-tall vectors and quadratic polynomials form vector spaces, as the sum and scalar multiple of vectors in these spaces maintain closure, demonstrating the generalizability of vector space concepts.']}, {'end': 12784.386, 'start': 12291.053, 'title': 'Vector spaces and linear combinations', 'summary': 'Discusses the concept of vector spaces, emphasizing the ability to take linear combinations of vectors and the closure property, illustrated through examples of quadratic polynomials and 3x3 matrices as vector spaces, and addresses trivial spaces, the origin of the 10 rules for vector spaces, and the technical manipulations to derive certain rules.', 'duration': 493.333, 'highlights': ['Vector spaces are about taking linear combinations and having closure under linear combinations, demonstrated through examples of quadratic polynomials and 3x3 matrices as vector spaces.', 'Illustration of the trivial space, a vector space containing only the zero vector and the explanation of the 10 rules for vector spaces, including the derivation of some rules from others.', 'Explanation of the technical manipulations involved in deriving certain rules for vector spaces, such as showing that 0 times a vector equals the zero vector, and r times the zero vector is the zero vector.']}], 'duration': 1080.796, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI11703590.jpg', 'highlights': ['The sum of vectors in a plane through the origin results in a vector in the same plane, verified algebraically and geometrically, ensuring closure under addition.', 'The scalar multiple of a vector in a plane through the origin remains in the same plane, confirmed by checking for an arbitrary scalar, satisfying the closure under scalar multiplication.', 'The chapter explores different kinds of vector spaces, emphasizing the introduction of a plane through the origin in R3 represented by the equation 2x + y + 3z = 0.', 'Vector spaces are about taking linear combinations and having closure under linear combinations, demonstrated through examples of quadratic polynomials and 3x3 matrices as vector spaces.', 'Illustration of the trivial space, a vector space containing only the zero vector and the explanation of the 10 rules for vector spaces, including the derivation of some rules from others.']}, {'end': 13864.499, 'segs': [{'end': 12844.613, 'src': 'embed', 'start': 12785.046, 'weight': 0, 'content': [{'end': 12792.908, 'text': 'So last time, of course, and the time before, we looked at the definition of vector space, and we tried to get some sense of examples of vector space.', 'start': 12785.046, 'duration': 7.862}, {'end': 12796.629, 'text': 'So there are many different kinds of vector spaces two tall vectors.', 'start': 12793.368, 'duration': 3.261}, {'end': 12797.749, 'text': 'three tall vectors.', 'start': 12796.629, 'duration': 1.12}, {'end': 12798.929, 'text': 'two by two matrices.', 'start': 12797.749, 'duration': 1.18}, {'end': 12800.13, 'text': 'three by three matrices.', 'start': 12798.929, 'duration': 1.201}, {'end': 12801.55, 'text': 'two by four matrices.', 'start': 12800.13, 'duration': 1.42}, {'end': 12802.51, 'text': 'quadratic polynomials.', 'start': 12801.55, 'duration': 0.96}, {'end': 12811.232, 'text': "There's a whole lot of vector spaces and we tried to see a number of examples and to check through the conditions and the definition to verify that they are all vector spaces.", 'start': 12802.53, 'duration': 8.702}, {'end': 12819.358, 'text': "I, a number of times, mentioned that condition one and condition six, the closure conditions, are the ones that we'll worry about most of the time.", 'start': 12812.132, 'duration': 7.226}, {'end': 12825.602, 'text': 'They prove to be the ones that are the trickiest, the ones that are kind of the key, the sort of central point of the matter.', 'start': 12819.398, 'duration': 6.204}, {'end': 12827.764, 'text': 'So, with the check-in today,', 'start': 12826.483, 'duration': 1.281}, {'end': 12835.43, 'text': "I want to reinforce a little bit what makes something a vector space and also give a little bit of a suggestion of what's coming.", 'start': 12827.764, 'duration': 7.666}, {'end': 12836.65, 'text': 'So here we go.', 'start': 12836.03, 'duration': 0.62}, {'end': 12840.111, 'text': 'It says this is a plane through the origin in R3.', 'start': 12836.67, 'duration': 3.441}, {'end': 12844.613, 'text': "And I want to verify that it's a verify.", 'start': 12840.572, 'duration': 4.041}], 'summary': 'The lecture discussed various examples of vector spaces and emphasized the importance of closure conditions.', 'duration': 59.567, 'max_score': 12785.046, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI12785046.jpg'}, {'end': 13408.757, 'src': 'embed', 'start': 13380.878, 'weight': 1, 'content': [{'end': 13383.961, 'text': "It's clearly a subset of the 2x2 matrices.", 'start': 13380.878, 'duration': 3.083}, {'end': 13391.89, 'text': "so it's going to be a subspace of the 2x2 matrices, and what we're finding is that vector spaces can have inside them vector spaces,", 'start': 13383.961, 'duration': 7.929}, {'end': 13393.171, 'text': 'and this is just an example.', 'start': 13391.89, 'duration': 1.281}, {'end': 13402.954, 'text': 'So again, one of the best ways to understand what something is is to give examples of it, not to contrast the examples of it, yes,', 'start': 13395.39, 'duration': 7.564}, {'end': 13404.035, 'text': 'with the examples of it not.', 'start': 13402.954, 'duration': 1.081}, {'end': 13407.276, 'text': 'You understand what a dog is in part by pointing to things that are not dog.', 'start': 13404.075, 'duration': 3.201}, {'end': 13408.757, 'text': "That's a cat, that's not a dog.", 'start': 13407.697, 'duration': 1.06}], 'summary': 'Vector spaces can contain subspaces, illustrated by 2x2 matrices.', 'duration': 27.879, 'max_score': 13380.878, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI13380878.jpg'}, {'end': 13501.939, 'src': 'embed', 'start': 13473.957, 'weight': 3, 'content': [{'end': 13479.924, 'text': "They say something that we've noticed many times in the few videos that we've been doing so far,", 'start': 13473.957, 'duration': 5.967}, {'end': 13482.907, 'text': "but we've noticed that it's about linear combinations.", 'start': 13479.924, 'duration': 2.983}, {'end': 13492.355, 'text': "So a subset of a vector space is itself a vector space provided that it's closed under linear combinations of pairs of vectors.", 'start': 13484.232, 'duration': 8.123}, {'end': 13495.176, 'text': "We've often taken linear combinations of pairs of vectors.", 'start': 13492.635, 'duration': 2.541}, {'end': 13501.939, 'text': "And also that's the same thing as saying you take linear combinations of any number of vectors, R1, S1 through Rn, Sn.", 'start': 13495.616, 'duration': 6.323}], 'summary': 'Linear combinations of vectors form a vector space if closed under pairs and any number of vectors.', 'duration': 27.982, 'max_score': 13473.957, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI13473957.jpg'}], 'start': 12785.046, 'title': 'Vector space and subspace verification', 'summary': 'Discusses the verification of a plane through the origin in r3 as a vector space, emphasizing closure under linear combinations and provides examples of subspaces within vector spaces, including quadratic polynomials and 2x2 matrices.', 'chapters': [{'end': 13037.031, 'start': 12785.046, 'title': 'Vector space verification', 'summary': 'Discusses the verification of a plane through the origin in r3 as a vector space by demonstrating the closure under linear combinations, illustrated through a specific example and emphasizing the concept of vector spaces being closed under linear combinations.', 'duration': 251.985, 'highlights': ['The chapter focuses on the verification of a plane through the origin in R3 as a vector space by demonstrating the closure under linear combinations, emphasizing the concept of vector spaces being closed under linear combinations.', 'The speaker mentions that condition one and condition six, the closure conditions, are the trickiest and central to the matter in verifying vector spaces.', 'Examples of different kinds of vector spaces, such as two tall vectors, three tall vectors, two by two matrices, three by three matrices, two by four matrices, and quadratic polynomials, are discussed, highlighting the variety of vector spaces.', 'The speaker provides an informal verification by demonstrating an arbitrary linear combination of two members of the space, emphasizing the closure property of vector spaces under linear combinations.', "An example involving the set of x y z's, where x + 2y + 3z = 0, is used to illustrate the verification process by taking an arbitrary linear combination of two vectors and showing that it results in a member of the space."]}, {'end': 13379.597, 'start': 13037.071, 'title': 'Linear combinations and subspaces', 'summary': 'Discusses linear combinations, subspaces, and vector spaces, emphasizing the concept of subspaces within vector spaces and providing examples of subspaces, including the collection of 2x2 matrices with specific limitations.', 'duration': 342.526, 'highlights': ["The chapter emphasizes the concept of subspaces within vector spaces, explaining that a subspace is a subset that's itself a vector space under the inherited operation, and provides examples of subspaces, including the collection of 2x2 matrices with specific limitations.", "The chapter discusses linear combinations and subspaces, highlighting that for any vector space, a subspace is a subset that's itself a vector space under the inherited operation, and introduces the concept of proper subspaces.", "The chapter provides examples of subspaces, including the collection of 2x2 matrices with specific limitations and emphasizes that a subspace is a subset that's itself a vector space under the inherited operation."]}, {'end': 13864.499, 'start': 13380.878, 'title': 'Subspaces and vector spaces', 'summary': 'Discusses the concept of subspaces within vector spaces, emphasizing their closure under linear combinations, and provides examples of subspaces within quadratic polynomials.', 'duration': 483.621, 'highlights': ['Subspaces are vector spaces that can exist within other vector spaces, and their closure under linear combinations is a key concept.', 'The concept of closure under linear combinations is a key idea for vector spaces and subspaces.', 'Examples of subspaces within quadratic polynomials, such as those with identical coefficients, are provided.']}], 'duration': 1079.453, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI12785046.jpg', 'highlights': ['The chapter focuses on the verification of a plane through the origin in R3 as a vector space by demonstrating the closure under linear combinations, emphasizing the concept of vector spaces being closed under linear combinations.', "The chapter emphasizes the concept of subspaces within vector spaces, explaining that a subspace is a subset that's itself a vector space under the inherited operation, and provides examples of subspaces, including the collection of 2x2 matrices with specific limitations.", 'Examples of different kinds of vector spaces, such as two tall vectors, three tall vectors, two by two matrices, three by three matrices, two by four matrices, and quadratic polynomials, are discussed, highlighting the variety of vector spaces.', 'Subspaces are vector spaces that can exist within other vector spaces, and their closure under linear combinations is a key concept.', 'The speaker mentions that condition one and condition six, the closure conditions, are the trickiest and central to the matter in verifying vector spaces.']}, {'end': 15254.388, 'segs': [{'end': 13933.546, 'src': 'embed', 'start': 13896.311, 'weight': 0, 'content': [{'end': 13902.613, 'text': "That parameterization proves to be a very convenient way to understand vector spaces, and we'll talk about that more as we move through.", 'start': 13896.311, 'duration': 6.302}, {'end': 13904.274, 'text': 'but just giving you a little preview.', 'start': 13902.613, 'duration': 1.661}, {'end': 13909.136, 'text': "Okay, and here's another example.", 'start': 13907.596, 'duration': 1.54}, {'end': 13911.197, 'text': 'This subspace is a plane.', 'start': 13909.356, 'duration': 1.841}, {'end': 13912.398, 'text': "It's a plane through the origin.", 'start': 13911.257, 'duration': 1.141}, {'end': 13923.458, 'text': "I want to illustrate that you can also understand it as parameterized, and then that makes it easy to show that it's closed under linear combinations.", 'start': 13913.43, 'duration': 10.028}, {'end': 13933.546, 'text': 'So to parameterize the plane, I think of 2x minus y plus z equals zero as a linear system that has only one equation.', 'start': 13925.099, 'duration': 8.447}], 'summary': 'Parameterization simplifies understanding vector spaces, illustrated with an example of a subspace as a plane, making it easy to show closure under linear combinations.', 'duration': 37.235, 'max_score': 13896.311, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI13896311.jpg'}, {'end': 14329.047, 'src': 'embed', 'start': 14302.659, 'weight': 3, 'content': [{'end': 14309.561, 'text': "So, for today's check-in, we're thinking about subspaces and we're thinking about a particular technique we saw with subspaces,", 'start': 14302.659, 'duration': 6.902}, {'end': 14311.782, 'text': 'and that is the parameterization technique.', 'start': 14309.561, 'duration': 2.221}, {'end': 14317.964, 'text': "I'm going to do that twice, and then I'm going to briefly mention why the parameterization shows that something is a subspace.", 'start': 14312.282, 'duration': 5.682}, {'end': 14324.366, 'text': 'Okay, so this says this is a subset of the space of quadratic polynomials.', 'start': 14319.504, 'duration': 4.862}, {'end': 14329.047, 'text': "so I'm going to refer to members of this subset as vectors because they're members of a vector space.", 'start': 14324.366, 'duration': 4.681}], 'summary': 'Discussion on subspaces and parameterization technique for quadratic polynomials.', 'duration': 26.388, 'max_score': 14302.659, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI14302659.jpg'}, {'end': 14893.397, 'src': 'embed', 'start': 14864.857, 'weight': 4, 'content': [{'end': 14867.898, 'text': "is that if it's closed under linear combinations, then it's a subspace.", 'start': 14864.857, 'duration': 3.041}, {'end': 14869.398, 'text': 'Okay, very good.', 'start': 14868.758, 'duration': 0.64}, {'end': 14874.62, 'text': "Okay, so we're ready to talk then about collections of linear combinations.", 'start': 14870.439, 'duration': 4.181}, {'end': 14883.914, 'text': "We've come to see that subspaces, vector spaces in general, are understood are well understood as a collection of linear combinations.", 'start': 14874.8, 'duration': 9.114}, {'end': 14887.175, 'text': 'You take a free collection where the letters are unrestricted.', 'start': 14883.954, 'duration': 3.221}, {'end': 14889.296, 'text': "It doesn't say b plus 2c equals 0.", 'start': 14887.195, 'duration': 2.101}, {'end': 14893.397, 'text': "It just simply says all b's and all c's kind of stuff.", 'start': 14889.296, 'duration': 4.101}], 'summary': 'Subspaces are understood as collections of unrestricted linear combinations.', 'duration': 28.54, 'max_score': 14864.857, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI14864857.jpg'}, {'end': 15196.39, 'src': 'embed', 'start': 15170.523, 'weight': 1, 'content': [{'end': 15176.245, 'text': 'then a linear combination of those linear combinations is again a linear combination and so is an element to the span.', 'start': 15170.523, 'duration': 5.722}, {'end': 15184.687, 'text': "So spans are a particularly felicitous way to view subspaces or spaces in general, since anything's a subspace in itself.", 'start': 15177.305, 'duration': 7.382}, {'end': 15193.629, 'text': "So just to illustrate that previous lemma, sometimes an illustration helps a person more than going through the i's and j's.", 'start': 15187.728, 'duration': 5.901}, {'end': 15196.39, 'text': "Not that the i's and j's are hard, but just sometimes it's more helpful.", 'start': 15193.689, 'duration': 2.701}], 'summary': "Spans are a felicitous way to view subspaces or spaces, as anything's a subspace in itself.", 'duration': 25.867, 'max_score': 15170.523, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI15170523.jpg'}], 'start': 13865.519, 'title': 'Understanding subspaces and parameterization', 'summary': 'Emphasizes the significance of parameterization in understanding vector spaces and subspaces, highlighting the convenience of parameterization in showing closure under linear combinations. it also covers the concept of subspaces in linear algebra and demonstrates the parameterization technique to show that a set is a subspace, using specific examples and explanations. additionally, the chapter discusses understanding subspaces as collections of unrestricted linear combinations and provides examples of vector spaces and subspaces, including the span of one element set, the span of a collection of vectors in r3, and the span of two specific vectors forming the xy-plane.', 'chapters': [{'end': 14140.536, 'start': 13865.519, 'title': 'Parameterization of subspaces', 'summary': 'Emphasizes the significance of parameterization in understanding vector spaces and subspaces, highlighting the convenience of parameterization in showing closure under linear combinations and the importance of linearity. it also illustrates how parameterization facilitates the description of subspaces and the determination of linear combinations.', 'duration': 275.017, 'highlights': ['The parameterization of subspaces proves to be a very convenient way to understand vector spaces, showcasing its significance in demonstrating closure under linear combinations and providing a preview of its importance in the study of subspaces.', 'Illustrating the parameterization of a plane as a linear system with one equation and expressing the leading variables in terms of free variables, showcasing the practical application of parameterization in understanding subspaces.', 'Demonstrating the ease of showing closure under linear combinations through parameterization and emphasizing the equivalence of different descriptions of vectors using parameterization, highlighting its utility in understanding subspaces.', 'Emphasizing the convenience of parameterization in showing that a linear combination of two vectors results in a third vector of the same form, showcasing its role in demonstrating linearity and its importance in understanding subspaces.', 'Highlighting the significance of parameterization in explicitly representing members of a subspace as a linear combination and emphasizing the linearity of linear combinations of linear combinations, ultimately serving as a key to understanding subspaces and vector spaces.']}, {'end': 14685.394, 'start': 14141.809, 'title': 'Subspaces and parameterization', 'summary': 'Discusses the concept of subspaces in linear algebra and demonstrates the parameterization technique to show that a set is a subspace, using specific examples and explanations.', 'duration': 543.585, 'highlights': ['The upper triangle of the 2x2 matrices are all the same number, while the bottom part (D) can vary.', 'Parameterizing the set to show it is a subspace involves expressing its members as a linear combination, demonstrating closure under linear combinations.', 'Demonstrating the technique of parameterization with the two parameters a and c to provide a particularly handy description of a vector space.', 'The chapter emphasizes the use of the parameterization technique to understand and describe vector spaces, with a focus on quadratic polynomials as vectors.', 'Illustrating the process of parameterization by providing specific examples and non-examples, showcasing the importance of the technique in understanding the properties of vector spaces.']}, {'end': 14960.934, 'start': 14685.734, 'title': 'Understanding subspaces as linear combinations', 'summary': 'Covers understanding subspaces as collections of unrestricted linear combinations, with a key result being that closure under linear combinations makes something a subspace, with examples and explanations provided.', 'duration': 275.2, 'highlights': ['Understanding subspaces as a collection of linear combinations', 'Closure under linear combinations makes something a subspace', 'Parameterizing to obtain a subspace']}, {'end': 15254.388, 'start': 14962.835, 'title': 'Vector spaces and subspaces', 'summary': 'Discusses examples of vector spaces and subspaces, including the span of one element set, the span of a collection of vectors in r3, and the span of two specific vectors forming the xy-plane, illustrating the concept of subspaces and spans in a vector space.', 'duration': 291.553, 'highlights': ['The span of a collection of vectors in R3 forms a subspace, such as the xy-plane, by taking linear combinations of the vectors to generate the span, as illustrated by solving a linear system to find a unique solution, facilitating the understanding of subspace formation and span in a vector space.', 'The lemma states that in a vector space, the span of any subset is a subspace, and when taking linear combinations of members of the span, the result stays inside the span, providing a felicitous way to view subspaces or spaces in general.', "Illustrating the concept of spans and subspaces with examples of linear combinations of specific vectors in a vector space, simplifying the understanding of abstract concepts for learners who may find working with i's and j's challenging, emphasizing the practical approach to understanding vector spaces and subspaces."]}], 'duration': 1388.869, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI13865519.jpg', 'highlights': ['The parameterization of subspaces proves to be a very convenient way to understand vector spaces, showcasing its significance in demonstrating closure under linear combinations and providing a preview of its importance in the study of subspaces.', 'The span of a collection of vectors in R3 forms a subspace, such as the xy-plane, by taking linear combinations of the vectors to generate the span, as illustrated by solving a linear system to find a unique solution, facilitating the understanding of subspace formation and span in a vector space.', 'Illustrating the parameterization of a plane as a linear system with one equation and expressing the leading variables in terms of free variables, showcasing the practical application of parameterization in understanding subspaces.', 'The chapter emphasizes the use of the parameterization technique to understand and describe vector spaces, with a focus on quadratic polynomials as vectors.', 'Understanding subspaces as a collection of linear combinations']}, {'end': 16390.614, 'segs': [{'end': 15380.591, 'src': 'embed', 'start': 15352.164, 'weight': 0, 'content': [{'end': 15356.325, 'text': 'I connected the xy-plane to R3 because the xy-plane is a subspace.', 'start': 15352.164, 'duration': 4.161}, {'end': 15357.906, 'text': "It's the span of s-hat.", 'start': 15356.765, 'duration': 1.141}, {'end': 15360.506, 'text': 'Anyway, the xy-plane is a subspace of R3.', 'start': 15357.986, 'duration': 2.52}, {'end': 15362.667, 'text': "So that's what the lines are showing.", 'start': 15361.006, 'duration': 1.661}, {'end': 15371.985, 'text': 'And the other thing a person notices about this picture is that the subspaces fall naturally into into layers.', 'start': 15365.007, 'duration': 6.978}, {'end': 15377.269, 'text': "There's three dimensionals, there's two dimensionals, there's one dimensionals, and there's zero dimensionals.", 'start': 15372.105, 'duration': 5.164}, {'end': 15380.591, 'text': "I'm going to call that zero dimensionals because it's the span of the empty set.", 'start': 15377.409, 'duration': 3.182}], 'summary': 'The xy-plane is a subspace of r3, and subspaces naturally fall into layers of dimensions: 3, 2, 1, and 0.', 'duration': 28.427, 'max_score': 15352.164, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI15352164.jpg'}, {'end': 15674.073, 'src': 'embed', 'start': 15648.327, 'weight': 3, 'content': [{'end': 15654.289, 'text': "What do I mean by how much freedom you have? Well, we're going to make that precise by parameterizing.", 'start': 15648.327, 'duration': 5.962}, {'end': 15658.143, 'text': 'The top layer and the bottom layer are not especially hard.', 'start': 15655.421, 'duration': 2.722}, {'end': 15664.907, 'text': "You can describe members of the quadratic polynomials by taking three different I'm going to call them vectors,", 'start': 15658.383, 'duration': 6.524}, {'end': 15671.011, 'text': "because they're members of a vector space three different vectors x, squared, x and 1..", 'start': 15664.907, 'duration': 6.104}, {'end': 15674.073, 'text': 'And you can describe members of the trivial subspace.', 'start': 15671.011, 'duration': 3.062}], 'summary': 'Freedom quantified by parameterizing; describes quadratic polynomials by 3 vectors x, x^2, 1.', 'duration': 25.746, 'max_score': 15648.327, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI15648327.jpg'}, {'end': 15801.309, 'src': 'embed', 'start': 15774.313, 'weight': 1, 'content': [{'end': 15784.68, 'text': "What we're seeing here is that quadratic polynomials, real three-space, four-by-four matrices, whatever it is we look at,", 'start': 15774.313, 'duration': 10.367}, {'end': 15788.403, 'text': 'vector spaces are most naturally described as spans.', 'start': 15784.68, 'duration': 3.723}, {'end': 15794.767, 'text': 'We get a description of something as a span by parameterizing.', 'start': 15790.264, 'duration': 4.503}, {'end': 15801.309, 'text': "It will be the case that we're interested in the number of members of the spanning set.", 'start': 15796.184, 'duration': 5.125}], 'summary': 'Vector spaces are best described as spans, with interest in the number of members in the spanning set.', 'duration': 26.996, 'max_score': 15774.313, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI15774313.jpg'}, {'end': 15858.516, 'src': 'embed', 'start': 15827.056, 'weight': 4, 'content': [{'end': 15834.863, 'text': 'Okay, so last time we saw that you can best describe vector spaces subspaces or spaces in general,', 'start': 15827.056, 'duration': 7.807}, {'end': 15839.246, 'text': 'best describe vector spaces as a span of a collection of members.', 'start': 15834.863, 'duration': 4.383}, {'end': 15843.77, 'text': 'So all linear combinations of a collection of members, which is called span.', 'start': 15839.647, 'duration': 4.123}, {'end': 15847.554, 'text': 'So the check-in here today is trying to illustrate that.', 'start': 15844.331, 'duration': 3.223}, {'end': 15858.516, 'text': 'important idea is that you can, you can, You can express any member of that space, the collection of a plus bx plus cx squared,', 'start': 15847.554, 'duration': 10.962}], 'summary': 'Vector spaces can be best described as the span of a collection of members, allowing for the expression of any member through linear combinations.', 'duration': 31.46, 'max_score': 15827.056, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI15827056.jpg'}], 'start': 15255.632, 'title': 'Organizing subspaces and vector spaces', 'summary': 'Illustrates the organization of subspaces by layers of dimensions and explains vector spaces as a span of a collection of members, showcasing natural categorization based on the number of parameters used to describe them.', 'chapters': [{'end': 15467.869, 'start': 15255.632, 'title': 'Illustrating order in subspaces', 'summary': 'Illustrates the order and organization of subspaces, showcasing how subspaces are organized into layers of three dimensionals, two dimensionals, one dimensionals, and zero dimensionals, and explains how subspaces are parametrized.', 'duration': 212.237, 'highlights': ['The subspaces fall naturally into layers of three dimensionals, two dimensionals, one dimensionals, and zero dimensionals, showcasing the order and organization of subspaces.', 'Subspaces are parametrized to be described by a certain number of vectors; all the planes are parametrized to be described by two vectors, all the lines by one vector, and so on.', 'The xy-plane is a subspace of R3 and is the span of s-hat, showcasing the relationship between subspaces.', 'The trivial subspace, just the origin, is a zero-dimensional subspace and is connected to other subspaces to illustrate their relationships.']}, {'end': 15826.255, 'start': 15469.871, 'title': 'Subspace categorization in r3 and p2', 'summary': 'Illustrates how subspaces in r3 and p2 fall naturally into categories based on the number of vectors used to parametrize them, with r3 having subspaces described by three, two, one, and zero vectors, and p2 having subspaces described by three, two, one, and zero vectors, showcasing the natural categorization of vector spaces into layers based on the number of parameters used to describe them.', 'duration': 356.384, 'highlights': ['The subspaces in R3 and P2 fall naturally into categories, with R3 having subspaces described by three, two, one, and zero vectors, and P2 having subspaces described by three, two, one, and zero vectors, showcasing the natural categorization of vector spaces into layers based on the number of parameters used to describe them.', 'Quadratic polynomials in P2 are categorized into layers based on the number of parameters used to describe them, with the top layer being described by three vectors, the middle layer by two vectors, the next layer by one vector, and the bottom layer by zero vectors.', 'Vector spaces are most naturally described as spans, and a description of something as a span is attained by parameterizing, which is crucial in illustrating the structure and determining the number of members of the spanning set.', 'The number of members of the spanning set turns out to be an important number as it demonstrates the structure of the vector space and the limitations in the possible number of members in the spanning set.']}, {'end': 16390.614, 'start': 15827.056, 'title': 'Vector spaces and spanning sets', 'summary': 'Illustrates the concept of vector spaces as a span of a collection of members, including expressing any member of the space as a combination of members of a set, and the observation that the third member of the set is a combination of the other two.', 'duration': 563.558, 'highlights': ['The concept of vector spaces is best described as a span of a collection of members, allowing any member of the space to be expressed as a combination of members of a set.', 'The third member of the set is observed to be a combination of the first two members, demonstrating that the spanning set can be made even smaller.', "The process of expressing any member of the space as a combination of the set's members is illustrated through specific examples, showing the calculation and combination of coefficients to achieve the desired expression."]}], 'duration': 1134.982, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI15255632.jpg', 'highlights': ['Subspaces fall into layers of three, two, one, and zero dimensionals', 'Vector spaces are most naturally described as spans', 'The xy-plane is a subspace of R3 and is the span of s-hat', 'Quadratic polynomials in P2 are categorized into layers based on the number of parameters used to describe them', 'The number of members of the spanning set demonstrates the structure of the vector space']}, {'end': 19446.4, 'segs': [{'end': 17121.346, 'src': 'embed', 'start': 17095.454, 'weight': 0, 'content': [{'end': 17103.396, 'text': "We're interested here in figuring out when these systems have more than just a trivial solution.", 'start': 17095.454, 'duration': 7.942}, {'end': 17104.576, 'text': "That's linear dependence.", 'start': 17103.456, 'duration': 1.12}, {'end': 17107.417, 'text': "When they have only the trivial solution, that's linear independence.", 'start': 17104.855, 'duration': 2.562}, {'end': 17109.158, 'text': "And we'll talk more about it next time.", 'start': 17107.877, 'duration': 1.281}, {'end': 17110.259, 'text': 'Okay, very good.', 'start': 17109.738, 'duration': 0.521}, {'end': 17113.421, 'text': 'Bye-bye Hello, hi.', 'start': 17110.439, 'duration': 2.982}, {'end': 17116.543, 'text': "So we're talking about linear independence today.", 'start': 17114.061, 'duration': 2.482}, {'end': 17121.346, 'text': 'Last time we did the first linear independence part, and this is the second linear independence part.', 'start': 17116.803, 'duration': 4.543}], 'summary': 'Exploring linear dependence vs. independence in systems, continuing from previous discussion.', 'duration': 25.892, 'max_score': 17095.454, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI17095454.jpg'}, {'end': 17753.872, 'src': 'embed', 'start': 17722.108, 'weight': 3, 'content': [{'end': 17727.211, 'text': 'a set is linearly independent if, and only if, for any vector in S, its removal shrinks the span.', 'start': 17722.108, 'duration': 5.103}, {'end': 17732.855, 'text': "So if I take, for example, since it's here on the screen, I'll use it, 1, 2, 3, 4, 5, 6, that's linearly independent.", 'start': 17727.912, 'duration': 4.943}, {'end': 17739.547, 'text': 'If you throw out the 4, 5, 6, no longer do you span a plane.', 'start': 17735.785, 'duration': 3.762}, {'end': 17745.509, 'text': 'Now you only span a line, all the multiples of 1, 2, 3.', 'start': 17739.627, 'duration': 5.882}, {'end': 17746.509, 'text': "Here's another example.", 'start': 17745.509, 'duration': 1}, {'end': 17753.872, 'text': 'Of course, this is a linearly independent subset of the cubic polynomials 1 plus x, 1 minus x, and x squared.', 'start': 17746.549, 'duration': 7.323}], 'summary': 'Linear independence: removal of vector shrinks span. example: 1, 2, 3, 4, 5, 6.', 'duration': 31.764, 'max_score': 17722.108, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI17722108.jpg'}, {'end': 17965.29, 'src': 'embed', 'start': 17935.185, 'weight': 1, 'content': [{'end': 17940.748, 'text': 'So I took a set with many, many vectors in it, and clearly linearly dependent.', 'start': 17935.185, 'duration': 5.563}, {'end': 17942.849, 'text': 'For example, the second one is 3 halves the first.', 'start': 17940.788, 'duration': 2.061}, {'end': 17950.297, 'text': 'If I write down a linear relationship, I get a lot of R1s, R2s, and R3s.', 'start': 17944.45, 'duration': 5.847}, {'end': 17957.764, 'text': "Do Gauss's method, and you spot that R1 and R3 are leading variables, and the other aren't leading variables.", 'start': 17950.938, 'duration': 6.826}, {'end': 17959.545, 'text': 'Free variables.', 'start': 17958.844, 'duration': 0.701}, {'end': 17965.29, 'text': 'When you do the parameterization, there you see the R2, the R4, and the R5 are free variables.', 'start': 17960.566, 'duration': 4.724}], 'summary': "A set of many vectors was found to be linearly dependent, and the gauss's method identified free variables.", 'duration': 30.105, 'max_score': 17935.185, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI17935185.jpg'}, {'end': 18125.377, 'src': 'embed', 'start': 18098.777, 'weight': 2, 'content': [{'end': 18102.199, 'text': 'At that point, the vector you just added must be dependent on the previous ones.', 'start': 18098.777, 'duration': 3.422}, {'end': 18103.14, 'text': "That's all.", 'start': 18102.579, 'duration': 0.561}, {'end': 18104.761, 'text': 'So a small observation.', 'start': 18103.46, 'duration': 1.301}, {'end': 18115.391, 'text': "So we're now at the conclusion that linear independence and span interact in some way.", 'start': 18107.687, 'duration': 7.704}, {'end': 18125.377, 'text': 'In particular, a subset of a linearly independent set is also linearly independent, and a superset of a linearly dependent is also linearly dependent.', 'start': 18115.411, 'duration': 9.966}], 'summary': 'Linear independence and span are related; subsets and supersets have corresponding properties.', 'duration': 26.6, 'max_score': 18098.777, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI18098777.jpg'}, {'end': 19149.713, 'src': 'embed', 'start': 19124.815, 'weight': 4, 'content': [{'end': 19131.18, 'text': 'Okay, so basis is important for many reasons, but one reason that a basis is important is this theorem.', 'start': 19124.815, 'duration': 6.365}, {'end': 19141.107, 'text': 'In any vector space, a subset is a basis if and only if each vector in the space can be expressed as a linear combination of elements in the subset.', 'start': 19133.301, 'duration': 7.806}, {'end': 19149.713, 'text': 'Now before I go on, all that says so far, every element in the space can be expressed as a linear combination of elements in the subset.', 'start': 19141.227, 'duration': 8.486}], 'summary': "Basis is crucial; in a vector space, a subset is a basis if each vector can be expressed as a linear combination of the subset's elements.", 'duration': 24.898, 'max_score': 19124.815, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI19124815.jpg'}], 'start': 16390.994, 'title': 'Linear independence and basis in vector space', 'summary': 'Explains the concepts of linear independence and dependence, relationship between linear independence and span, interaction between linear independence and span, and the introduction to basis in vector space. it emphasizes the importance of having only trivial solutions to determine linear independence and the relevance of basis in various vector spaces.', 'chapters': [{'end': 17377.728, 'start': 16390.994, 'title': 'Linear independence concept', 'summary': "Explains the concept of linear independence and dependence, illustrating how to determine linear independence of a set of vectors through examples and gauss's method, emphasizing the importance of having only trivial solutions to determine linear independence.", 'duration': 986.734, 'highlights': ['The chapter introduces the concept of linear independence and dependence, emphasizing the importance of having only trivial solutions to determine linear independence.', "The chapter illustrates the process of determining linear independence of a set of vectors through examples and Gauss's method.", "The chapter provides examples of determining linear independence, emphasizing the use of Gauss's method and the importance of having only trivial solutions."]}, {'end': 18056.908, 'start': 17378.749, 'title': 'Relationship between linear independence and span', 'summary': 'Discusses the relationship between linear independence and span, illustrating that adding a vector to a set either leaves the span unchanged or increases it, and emphasizing the interaction between span and linear independence.', 'duration': 678.159, 'highlights': ['The relationship between linear independence and span is illustrated by the fact that adding a vector to a set either leaves the span unchanged or increases it, demonstrating the interaction between span and linear independence.', 'A set is linearly independent if none of its elements is a linear combination of the others, and the removal of any vector from a linearly independent set shrinks the span.', 'If a vector is in the span of a linearly independent set, adding it to the set will cause the new set to be linearly dependent, while adding a genuinely new vector will retain linear independence.']}, {'end': 18735.357, 'start': 18059.544, 'title': 'Linear independence and span interaction', 'summary': 'Explores the relationship between linear independence and span, explaining the interaction between the two concepts and demonstrating how adding elements to a set affects linear independence, ultimately determining whether the resulting set is linearly independent or dependent.', 'duration': 675.813, 'highlights': ['The check-in focuses on the interaction between linear independence and span, and demonstrates how adding elements to a set affects linear independence, ultimately determining whether the resulting set is linearly independent or dependent, with examples and calculations. (Relevance: 5)', 'The chapter establishes the criteria for a subset to be linearly dependent and explores the interaction between linear independence and span, highlighting the relationships between linearly independent and linearly dependent subsets. (Relevance: 4)', 'The chapter discusses the relationship between linear independence and span, emphasizing that a subset of a linearly independent set is also linearly independent, and a superset of a linearly dependent set is also linearly dependent, providing key insights into the interplay between linear independence and span. (Relevance: 3)', 'The process of adding new vectors to a linearly independent set is explored, revealing that if new vectors are already in the span, the set becomes dependent, whereas if the new vectors are not in the span, the set remains independent, providing valuable insights into the impact of vector addition on linear independence. (Relevance: 2)', 'The chapter presents the process of determining a spanning set for a plane, confirming its linear independence, and subsequently adding a vector to the spanning set to analyze whether the resulting set is linearly independent or dependent, demonstrating practical applications of the concepts discussed. (Relevance: 1)']}, {'end': 19067.013, 'start': 18735.357, 'title': 'Introduction to basis in vector space', 'summary': 'Introduces the concept of a basis as a sequence of vectors that is both linearly independent and spans the space, with examples demonstrating the properties and relevance of basis in various vector spaces.', 'duration': 331.656, 'highlights': ["A basis for vector space is a sequence of vectors that's both linearly independent and spans the space.", 'Example of a basis for the plane, 1 minus 1 and 1, 1, is provided, demonstrating its linear independence and spanning properties.', 'Example of a basis for the vector space of linear polynomials, 1 plus x and 1 minus x, is given, highlighting its linear independence and spanning properties.']}, {'end': 19446.4, 'start': 19067.901, 'title': 'Understanding vector space bases', 'summary': 'Explains the importance of a basis in a vector space, highlighting that a basis is a set of vectors that spans the space and is linearly independent, giving each vector in the space a unique way of being expressed as a linear combination of elements in the subset. it emphasizes the tension between spanning sets, which tend to be big, and linearly independent sets, which tend to be small, and how the intersection of these sets is a sweet spot.', 'duration': 378.499, 'highlights': ['The importance of a basis in a vector space is highlighted, emphasizing that a basis is a set of vectors that spans the space and is linearly independent, giving each vector in the space a unique way of being expressed as a linear combination of elements in the subset.', 'The tension between spanning sets, which tend to be big, and linearly independent sets, which tend to be small, is explained, with the intersection of these sets being a sweet spot.', 'Emphasizing the theorem that in a vector space, a subset is a basis if and only if each vector in the space can be expressed as a linear combination of elements in the subset in exactly one way.']}], 'duration': 3055.406, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI16390993.jpg', 'highlights': ['The chapter emphasizes the importance of having only trivial solutions to determine linear independence.', "The chapter illustrates the process of determining linear independence of a set of vectors through examples and Gauss's method.", 'The relationship between linear independence and span is illustrated by the fact that adding a vector to a set either leaves the span unchanged or increases it, demonstrating the interaction between span and linear independence.', 'A set is linearly independent if none of its elements is a linear combination of the others, and the removal of any vector from a linearly independent set shrinks the span.', 'The importance of a basis in a vector space is highlighted, emphasizing that a basis is a set of vectors that spans the space and is linearly independent, giving each vector in the space a unique way of being expressed as a linear combination of elements in the subset.']}, {'end': 20809.521, 'segs': [{'end': 19742.854, 'src': 'embed', 'start': 19702.574, 'weight': 0, 'content': [{'end': 19703.394, 'text': "I'm practically out.", 'start': 19702.574, 'duration': 0.82}, {'end': 19705.115, 'text': "So let's see how to go.", 'start': 19703.494, 'duration': 1.621}, {'end': 19707.957, 'text': 'It says 2 minus x, 3 plus 2x.', 'start': 19705.295, 'duration': 2.662}, {'end': 19709.238, 'text': "So that's a basis.", 'start': 19708.337, 'duration': 0.901}, {'end': 19716.108, 'text': 'And then it says what combination gives minus 1? minus x.', 'start': 19709.538, 'duration': 6.57}, {'end': 19719.79, 'text': "And so once I show it's a basis, of course, I put in diamond brackets.", 'start': 19716.108, 'duration': 3.682}, {'end': 19721.27, 'text': "So let's give it a whack.", 'start': 19719.81, 'duration': 1.46}, {'end': 19727.192, 'text': "We're looking at to show linear independence for question number three.", 'start': 19723.411, 'duration': 3.781}, {'end': 19742.854, 'text': "To show linear independence, I take c1 times the first, 2 minus x plus c2 times the second, 3 plus 2x And I'm going to write down here 0.", 'start': 19727.872, 'duration': 14.982}], 'summary': 'Demonstrating linear independence using specific equations and variables in a basis.', 'duration': 40.28, 'max_score': 19702.574, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI19702574.jpg'}, {'end': 20163.198, 'src': 'embed', 'start': 20135.212, 'weight': 1, 'content': [{'end': 20141.237, 'text': 'each vector in the space can be expressed as a linear combination of elements of the subset in one and only one way.', 'start': 20135.212, 'duration': 6.025}, {'end': 20147.726, 'text': "One way is not surprising because it's a spanning set, so of course everybody's expressible.", 'start': 20143.463, 'duration': 4.263}, {'end': 20148.907, 'text': "That's the point of spanning.", 'start': 20147.826, 'duration': 1.081}, {'end': 20156.553, 'text': "But only one way, that's the thing in this theorem that makes a point, that you can't express it in two ways or three ways or infinitely many ways.", 'start': 20149.328, 'duration': 7.225}, {'end': 20157.314, 'text': 'Only one way.', 'start': 20156.713, 'duration': 0.601}, {'end': 20163.198, 'text': "So if you say to me, what are the coefficients? You're not asking an ambiguous question.", 'start': 20159.015, 'duration': 4.183}], 'summary': 'Each vector can be expressed as a linear combination uniquely.', 'duration': 27.986, 'max_score': 20135.212, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI20135212.jpg'}, {'end': 20264.321, 'src': 'embed', 'start': 20235.591, 'weight': 2, 'content': [{'end': 20243.953, 'text': 'the representation of the vector with respect to the basis is the column vector of coefficients that you use to express V.', 'start': 20235.591, 'duration': 8.362}, {'end': 20246.794, 'text': 'So you write those coefficients down vertically in a vector.', 'start': 20243.953, 'duration': 2.841}, {'end': 20253.731, 'text': 'representation of v with respect to b.', 'start': 20249.187, 'duration': 4.544}, {'end': 20261.478, 'text': "Really, both v and b are variables here, but for the kinds of problems that we'll typically do, especially at the beginning,", 'start': 20253.731, 'duration': 7.747}, {'end': 20264.321, 'text': "we'll hold b constant and maybe vary v.", 'start': 20261.478, 'duration': 2.843}], 'summary': 'Represents vector with respect to basis using column vector of coefficients.', 'duration': 28.73, 'max_score': 20235.591, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI20235591.jpg'}, {'end': 20575.947, 'src': 'embed', 'start': 20544.765, 'weight': 4, 'content': [{'end': 20551.549, 'text': "So for next time, of course, we're going to be talking about dimension, and we'll see you for that.", 'start': 20544.765, 'duration': 6.784}, {'end': 20552.389, 'text': 'Okay, bye-bye.', 'start': 20551.569, 'duration': 0.82}, {'end': 20557.156, 'text': 'Hi, hello.', 'start': 20556.155, 'duration': 1.001}, {'end': 20562.939, 'text': 'Well, last time we did the second half of bases and we represented vectors with respect to bases.', 'start': 20557.656, 'duration': 5.283}, {'end': 20566.801, 'text': 'so naturally, the check-in asks us to represent some vectors with respect to bases.', 'start': 20562.939, 'duration': 3.862}, {'end': 20575.947, 'text': "So here we have two different bases, b1 and b2, and we're going to represent a single vector, b, with respect to those two bases.", 'start': 20567.422, 'duration': 8.525}], 'summary': 'Next session will cover dimension and representing vectors with respect to bases.', 'duration': 31.182, 'max_score': 20544.765, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI20544765.jpg'}], 'start': 19446.4, 'title': 'Vector space and linear systems', 'summary': "Covers determining linear independence and span of vectors, solving linear systems using gauss's method, demonstrating uniqueness of expressing vectors as linear combinations, and representing vectors with respect to bases, using specific examples and calculations.", 'chapters': [{'end': 19770.452, 'start': 19446.4, 'title': 'Vector space basis and linear independence', 'summary': 'Discusses determining linear independence and span of a given set of vectors in a vector space, illustrating the process with specific examples and calculations.', 'duration': 324.052, 'highlights': ['The chapter discusses determining linear independence and span of a given set of vectors in a vector space, illustrating the process with specific examples and calculations.', 'The sequence 1, 1, 1 and 1, -1 is proven to be a linearly independent set in the given vector space through the process of solving linear systems and finding unique solutions.', 'The vector 3, -2 is shown to be a linear combination of the basis elements 1, 1 and 1, -1 with specific calculations to find the coefficients, demonstrating the concept of span in the vector space.', 'The set {2 - x, 3 + 2x} is verified to be a basis in the given vector space by demonstrating linear independence through solving linear systems of equations with polynomials, reinforcing the concept of span and linear independence.']}, {'end': 20098.259, 'start': 19771.993, 'title': 'Linear system and basis problem', 'summary': "Covers solving a linear system using gauss's method to determine linear independence, finding the coefficients to express a polynomial as a linear combination, and demonstrating a unique way to express vectors as a linear combination.", 'duration': 326.266, 'highlights': ["Solving linear system using Gauss's method to determine linear independence", 'Finding coefficients to express a polynomial as a linear combination', 'Demonstrating a unique way to express vectors as a linear combination']}, {'end': 20474.999, 'start': 20099.64, 'title': 'Vector space basis theorem', 'summary': 'Discusses the uniqueness of expressing vectors in a vector space as a linear combination of basis vectors, using examples and highlighting the importance of unique representations and the relationship between vectors and their representatives.', 'duration': 375.359, 'highlights': ['The theorem states that in any vector space, a subset is a basis if, and only if, each vector in the space can be expressed as a linear combination of elements of the subset in one and only one way, emphasizing the importance of unique representations.', 'The representation of a vector with respect to a basis is the column vector of coefficients used to express the vector, highlighting the practical application of unique representations in vector spaces.', 'A relationship holds among a set of vectors if, and only if, that relationship holds among the representatives of those vectors, underscoring the significance of representations in understanding linear combinations.']}, {'end': 20809.521, 'start': 20477.38, 'title': 'Representing vectors with respect to bases', 'summary': 'Discusses representing vectors with respect to bases using examples and linear systems, demonstrating the process of finding vector representations and emphasizing the use of brain power alongside traditional methods. it also introduces the concept of representing polynomials as vectors in a vector space.', 'duration': 332.141, 'highlights': ['The chapter discusses representing vectors with respect to bases using examples and linear systems, demonstrating the process of finding vector representations.', 'The chapter emphasizes the use of brain power alongside traditional methods, showcasing the value of critical thinking in solving problems.', 'The chapter introduces the concept of representing polynomials as vectors in a vector space, expanding the understanding of vectors to include polynomial representations.']}], 'duration': 1363.121, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI19446400.jpg', 'highlights': ['The set {2 - x, 3 + 2x} is verified to be a basis in the given vector space by demonstrating linear independence through solving linear systems of equations with polynomials, reinforcing the concept of span and linear independence.', 'The theorem states that in any vector space, a subset is a basis if, and only if, each vector in the space can be expressed as a linear combination of elements of the subset in one and only one way, emphasizing the importance of unique representations.', 'The representation of a vector with respect to a basis is the column vector of coefficients used to express the vector, highlighting the practical application of unique representations in vector spaces.', 'The chapter discusses determining linear independence and span of a given set of vectors in a vector space, illustrating the process with specific examples and calculations.', 'The chapter discusses representing vectors with respect to bases using examples and linear systems, demonstrating the process of finding vector representations.']}, {'end': 23405.66, 'segs': [{'end': 20836.333, 'src': 'embed', 'start': 20813.192, 'weight': 2, 'content': [{'end': 20821.498, 'text': 'So again, to write the representation with respect to the basis B2 of V, it will be different than this representation but still be a representation.', 'start': 20813.192, 'duration': 8.306}, {'end': 20827.349, 'text': "I've got to write it in the correct order, minus 2, 0, 3.", 'start': 20822.118, 'duration': 5.231}, {'end': 20834.632, 'text': "You can absolutely write, for example, a subscript b2 if you're in a context where sometimes there's one basis and sometimes another, but we often,", 'start': 20827.349, 'duration': 7.283}, {'end': 20836.333, 'text': "when we're working, often leave that off.", 'start': 20834.632, 'duration': 1.701}], 'summary': 'The representation with respect to basis b2 of v is -2, 0, 3.', 'duration': 23.141, 'max_score': 20813.192, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI20813192.jpg'}, {'end': 21416.621, 'src': 'embed', 'start': 21386.476, 'weight': 3, 'content': [{'end': 21393.258, 'text': 'The dimension of a vector space is the number of vectors in any of its bases, and they all have the same number, so that definition is not ambiguous.', 'start': 21386.476, 'duration': 6.782}, {'end': 21396.918, 'text': 'In particular, where does the word dimension come from??', 'start': 21393.758, 'duration': 3.16}, {'end': 21403.7, 'text': "Well, in the regular n-dimensional space, Rn, that's how many members are in the standard basis.", 'start': 21397.238, 'duration': 6.462}, {'end': 21405.4, 'text': "So that's how many members are in any basis.", 'start': 21403.84, 'duration': 1.56}, {'end': 21407.417, 'text': "So that's where the word dimension comes from.", 'start': 21405.856, 'duration': 1.561}, {'end': 21416.621, 'text': 'The quadratic polynomials has dimension 3 because I can name a basis that has three elements, and any basis has the same number of elements.', 'start': 21409.338, 'duration': 7.283}], 'summary': 'The dimension of a vector space is the number of vectors in any of its bases, and they all have the same number, so that definition is not ambiguous.', 'duration': 30.145, 'max_score': 21386.476, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI21386476.jpg'}, {'end': 21493.327, 'src': 'embed', 'start': 21465.768, 'weight': 4, 'content': [{'end': 21468.329, 'text': 'So the solution set is a vector space of dimension two.', 'start': 21465.768, 'duration': 2.561}, {'end': 21469.25, 'text': "It's a plane.", 'start': 21468.509, 'duration': 0.741}, {'end': 21477.656, 'text': 'Because there are four letters here, this is a plane in R4, plane to the origin, in R4 spanned by those two vectors.', 'start': 21469.75, 'duration': 7.906}, {'end': 21486.502, 'text': 'Okay, a couple of results that the theorem that we just saw gives us sort of an immediate payoff.', 'start': 21480.378, 'duration': 6.124}, {'end': 21487.683, 'text': 'It gives us a number of results.', 'start': 21486.542, 'duration': 1.141}, {'end': 21493.327, 'text': 'So no linearly independent set can have a size greater than the dimension of the enclosing space.', 'start': 21488.243, 'duration': 5.084}], 'summary': "The solution set is a 2d vector space in r4, and no linearly independent set can exceed the enclosing space's dimension.", 'duration': 27.559, 'max_score': 21465.768, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI21465768.jpg'}, {'end': 22609.521, 'src': 'embed', 'start': 22583.74, 'weight': 0, 'content': [{'end': 22588.804, 'text': 'Next result, a very often used result, that the row rank and column rank are equal, very often used in practice.', 'start': 22583.74, 'duration': 5.064}, {'end': 22591.106, 'text': 'Row rank and column rank of a matrix are equal.', 'start': 22588.904, 'duration': 2.202}, {'end': 22598.552, 'text': 'The number of linearly independent rows and the number of linearly independent columns is the same number.', 'start': 22591.126, 'duration': 7.426}, {'end': 22605.097, 'text': 'And the proof is, at this point we have a lot of machinery, so the proof turns out to be not many sentences.', 'start': 22600.574, 'duration': 4.523}, {'end': 22609.521, 'text': 'You bring the matrix to reduced echelon form, not just echelon form, reduced echelon form.', 'start': 22605.698, 'duration': 3.823}], 'summary': 'Row rank equals column rank in a matrix, indicating equal linearly independent rows and columns, proven through reduced echelon form.', 'duration': 25.781, 'max_score': 22583.74, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI22583740.jpg'}, {'end': 22894.16, 'src': 'embed', 'start': 22866.247, 'weight': 1, 'content': [{'end': 22870.09, 'text': 'Any one of these statements, any one of these five statements, is the same as saying any of the others.', 'start': 22866.247, 'duration': 3.843}, {'end': 22876.354, 'text': "For example, if you say A is nonsingular, that's the same as saying the columns of A form a linearly independent set.", 'start': 22870.51, 'duration': 5.844}, {'end': 22877.675, 'text': "That's an if and only if statement.", 'start': 22876.395, 'duration': 1.28}, {'end': 22882.559, 'text': 'A is nonsingular if and only if the columns of A form a linearly independent set.', 'start': 22878.256, 'duration': 4.303}, {'end': 22886.262, 'text': 'So I have five statements, each of which is equivalent to the other.', 'start': 22883.74, 'duration': 2.522}, {'end': 22889.076, 'text': 'each of which is if and only if the other.', 'start': 22887.355, 'duration': 1.721}, {'end': 22894.16, 'text': 'Now you can go through and say prove one if and only if, two, one if and only if three.', 'start': 22889.737, 'duration': 4.423}], 'summary': 'Five statements are equivalent, forming a linearly independent set.', 'duration': 27.913, 'max_score': 22866.247, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI22866247.jpg'}], 'start': 20813.192, 'title': 'Vector space dimensionality', 'summary': 'Covers representation of vectors, finite dimensional vector spaces, vector space basis and dimension, n-dimensional spaces, preservation of column rank, and emphasizes the importance of homework and recommended videos for understanding the material.', 'chapters': [{'end': 20865.167, 'start': 20813.192, 'title': 'Basis and dimension', 'summary': 'Covers the representation of vectors with respect to different bases, discussing the idea of leaving off subscripts for basis representation and transitioning to the topic of dimensionality.', 'duration': 51.975, 'highlights': ['The section covers the representation of vectors with respect to different bases, emphasizing the importance of writing it in the correct order and the possibility of leaving off subscripts for basis representation.', 'The chapter transitions from discussing basis to introducing the topic of dimension, indicating an upcoming focus on dimensionality in the context of vector spaces.']}, {'end': 21111.497, 'start': 20865.587, 'title': 'Finite dimensional vector spaces', 'summary': 'Explains finite dimensional vector spaces, demonstrating that a vector space is finite dimensional if it has a basis with only finitely many vectors, and proving that for any finite dimensional space, all bases have the same number of elements.', 'duration': 245.91, 'highlights': ['A vector space is finite dimensional if it has a basis with only finitely many vectors.', 'P2 is finite dimensional because it has at least one basis with finitely many elements.', 'The theorem that in a finite dimensional vector space all bases have the same number of elements.', 'The proof in the book is a little more abstract, a little more conceptual.']}, {'end': 21673.005, 'start': 21113.418, 'title': 'Vector space basis and dimension', 'summary': 'Discusses the concept of vector space basis and dimension, emphasizing that all bases have the same minimal size, defining the dimension of a vector space, providing examples of dimensions for specific vector spaces, explaining the dimension of the solution set for a linear system, and highlighting the relationship between linearly independent sets and the dimension of the enclosing space.', 'duration': 559.587, 'highlights': ['The dimension of a vector space is the number of vectors in any of its bases, and they all have the same number, so that definition is not ambiguous.', "In a regular n-dimensional space, Rn, the dimension is the number of members in the standard basis, hence the word 'dimension'.", 'The solution set for a specific linear system is a vector space of dimension two, forming a plane in R4 spanned by two vectors.', 'No linearly independent set can have a size greater than the dimension of the enclosing space, reinforcing the intuition that linearly independent sets tend to be small.']}, {'end': 22514.331, 'start': 21675.867, 'title': 'Dimension and basis in linear algebra', 'summary': 'Discusses the concept of n-dimensional spaces, linear independence, spanning sets, dimension of vector spaces, row space, column space, and their relevance to linear systems in a comprehensive manner with examples, emphasizing the key concepts and connections between them.', 'duration': 838.464, 'highlights': ['The proof that a subset with n vectors is linearly independent if and only if it spans the space.', "The explanation of how Gauss's method produces a basis for the row space of a matrix.", 'The concept that row equivalent matrices have the same row space and row rank.', 'The method of finding the basis of a column space by transposing and applying Gaussian elimination.', 'The explanation of how the right-hand side of a linear system relates to the column space of the matrix.']}, {'end': 22982.265, 'start': 22514.651, 'title': 'Preservation of column rank', 'summary': 'Discusses how row operations preserve column rank, with the key point that row operations do not change the number of linearly unrelated columns, and the important result that the row rank and column rank of a matrix are equal, providing a practical illustration through an example.', 'duration': 467.614, 'highlights': ["The row rank and column rank of a matrix are equal, indicating the same number of linearly independent rows and columns, with the practical example illustrating the calculation of column rank through Gauss's method resulting in 3 linearly independent rows and hence a column rank of 3.", "The equivalence between the rank of a matrix and the dimension of the vector space of solutions of the associated homogeneous system, expressed as 'rank of a is r' and 'dimension n minus r', providing a clear relationship between the rank of the matrix and the degrees of freedom in the system.", 'The demonstration of the equivalence of five statements relating to a nonsingular matrix and the linear independence of its columns, emphasizing the interrelation between these properties and their implications for unique solutions in linear systems.']}, {'end': 23405.66, 'start': 22982.465, 'title': 'Importance of homework and recommended videos', 'summary': 'Underscores the importance of doing homework for understanding the material, emphasizes the value of recommended educational videos, and provides a straightforward check-in on row and column spaces.', 'duration': 423.195, 'highlights': ['The homework is emphasized as crucial for understanding the material, over watching videos, with the reminder that the whole point is to be able to answer questions about the systems.', 'Recommends educational videos by 3Blue1Brown on YouTube, noting their ability to provide additional insight and aid in understanding the material.', 'Provides a straightforward check-in on row and column spaces, demonstrating the calculation for finding bases and ranks for both spaces, and reaffirming the theorem that the row rank and the column rank are the same.']}], 'duration': 2592.468, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI20813192.jpg', 'highlights': ["The row rank and column rank of a matrix are equal, indicating the same number of linearly independent rows and columns, with the practical example illustrating the calculation of column rank through Gauss's method resulting in 3 linearly independent rows and hence a column rank of 3.", 'The demonstration of the equivalence of five statements relating to a nonsingular matrix and the linear independence of its columns, emphasizing the interrelation between these properties and their implications for unique solutions in linear systems.', 'The section covers the representation of vectors with respect to different bases, emphasizing the importance of writing it in the correct order and the possibility of leaving off subscripts for basis representation.', 'The dimension of a vector space is the number of vectors in any of its bases, and they all have the same number, so that definition is not ambiguous.', 'The solution set for a specific linear system is a vector space of dimension two, forming a plane in R4 spanned by two vectors.']}, {'end': 25818.398, 'segs': [{'end': 23661.206, 'src': 'embed', 'start': 23637.594, 'weight': 0, 'content': [{'end': 23646.38, 'text': 'And again very sensitive to the fact that for many folks taking this class the first time you see this definition will be like what the heck is this?', 'start': 23637.594, 'duration': 8.786}, {'end': 23653.062, 'text': "So you know we're going to do lots of examples, going to work through this at a deliberate pace.", 'start': 23647.18, 'duration': 5.882}, {'end': 23657.644, 'text': 'Isomorphism is the technical jargon here.', 'start': 23655.443, 'duration': 2.201}, {'end': 23661.206, 'text': 'Isomorphism between two vector spaces is a function.', 'start': 23657.785, 'duration': 3.421}], 'summary': 'Introduction to isomorphism and vector spaces.', 'duration': 23.612, 'max_score': 23637.594, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI23637594.jpg'}, {'end': 23771.517, 'src': 'embed', 'start': 23745.943, 'weight': 1, 'content': [{'end': 23752.727, 'text': 'But the very first thing is, how do I do homework question sort of number five? So that makes sense.', 'start': 23745.943, 'duration': 6.784}, {'end': 23757.01, 'text': 'So there are four things to show that a function is one-to-one, to show that a function is onto.', 'start': 23752.747, 'duration': 4.263}, {'end': 23761.493, 'text': 'to show that a function preserves addition, and to show that a function preserves scalar multiplications.', 'start': 23757.01, 'duration': 4.483}, {'end': 23765.873, 'text': 'Of course, this was labeled item number one on the previous slide, the first two.', 'start': 23762.23, 'duration': 3.643}, {'end': 23771.517, 'text': 'This was labeled item number two on the previous slide, the second two.', 'start': 23766.353, 'duration': 5.164}], 'summary': 'Key points include four criteria for function properties and their corresponding labels.', 'duration': 25.574, 'max_score': 23745.943, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI23745943.jpg'}, {'end': 24846.337, 'src': 'embed', 'start': 24822.711, 'weight': 3, 'content': [{'end': 24829.217, 'text': "Okay, so we've gone through a couple of these examples of showing that something is an isomorphism.", 'start': 24822.711, 'duration': 6.506}, {'end': 24834.041, 'text': 'So I think a person is ready to think about the homework.', 'start': 24829.257, 'duration': 4.784}, {'end': 24838.706, 'text': "And the next time, of course, we'll go through and look at some more for isomorphisms.", 'start': 24834.6, 'duration': 4.106}, {'end': 24839.888, 'text': 'Brand new idea.', 'start': 24839.027, 'duration': 0.861}, {'end': 24846.337, 'text': "We'll look at some more examples and try to kind of piece it all together and make some sense of what the parts have to say.", 'start': 24839.908, 'duration': 6.429}], 'summary': 'Exploring isomorphisms with examples and preparing for homework.', 'duration': 23.626, 'max_score': 24822.711, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI24822711.jpg'}, {'end': 24950.072, 'src': 'embed', 'start': 24918.675, 'weight': 2, 'content': [{'end': 24923.079, 'text': 'We saw last time the how-to, four different steps that a person needs to show.', 'start': 24918.675, 'duration': 4.404}, {'end': 24930.745, 'text': 'A person needs to show a one-to-one, onto, and preservation of addition and preservation of scale and multiplication.', 'start': 24923.099, 'duration': 7.646}, {'end': 24932.566, 'text': "So I'll just start in.", 'start': 24930.805, 'duration': 1.761}, {'end': 24934.888, 'text': 'I better remind myself how it goes.', 'start': 24932.586, 'duration': 2.302}, {'end': 24936.63, 'text': "Let's see.", 'start': 24936.129, 'duration': 0.501}, {'end': 24950.072, 'text': "It is a, b is associated with, and it's a minus b plus a plus b x.", 'start': 24936.93, 'duration': 13.142}], 'summary': 'Four steps for demonstrating mathematical properties were discussed.', 'duration': 31.397, 'max_score': 24918.675, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI24918675.jpg'}], 'start': 23405.66, 'title': 'Isomorphisms in linear algebra', 'summary': "Introduces isomorphisms, emphasizing one-to-one, onto functions preserving vector space structure. it covers techniques to show functions' properties and discusses preservation of addition and scalar multiplication with examples, verification steps, and mental modeling.", 'chapters': [{'end': 23745.903, 'start': 23405.66, 'title': 'Understanding isomorphisms in linear algebra', 'summary': 'Introduces the concept of isomorphisms between vector spaces, highlighting the key idea that isomorphism between two vector spaces is a function that is one-to-one, onto, and preserves the structure of the space, with examples illustrating the concept.', 'duration': 340.243, 'highlights': ['The chapter introduces the concept of isomorphisms between vector spaces and emphasizes the importance of examples and careful development in understanding the concept.', 'The key idea of isomorphism is the association between vector spaces that upholds throughout operations, such as addition and scalar multiplication.', 'The definition of isomorphism between two vector spaces is a function that is one-to-one, onto, and preserves the structure of the space, with a detailed explanation of verifying these properties.']}, {'end': 24065.409, 'start': 23745.943, 'title': 'Functions and vectors in linear algebra', 'summary': 'Covers the techniques to show a function is one-to-one, onto, preserves addition, and preserves scalar multiplication, with examples and emphasis on understanding through concrete examples and the technical aspects of function mapping.', 'duration': 319.466, 'highlights': ['The chapter covers the techniques to show a function is one-to-one, onto, preserves addition, and preserves scalar multiplication.', 'Emphasis on understanding through concrete examples and the technical aspects of function mapping.', 'Taking out a pencil and putting down some numbers to understand abstract concepts.']}, {'end': 24779.415, 'start': 24067.329, 'title': 'Preservation of addition and scalar multiplication', 'summary': 'Explains the preservation of addition and scalar multiplication, emphasizing the importance of understanding the steps and showing examples and verification steps for isomorphism. it also highlights the process of working through examples by writing down the beginning and end and making them match.', 'duration': 712.086, 'highlights': ['The chapter emphasizes the importance of understanding the steps and the concept of preservation when demonstrating the preservation of addition and scalar multiplication.', 'The speaker emphasizes the importance of working through examples by writing down the beginning and end and making them match, illustrating the process with specific examples and verification steps for isomorphism.', 'The speaker highlights the process of working through examples by writing down the beginning and end and making them match, emphasizing the importance of understanding the steps and showing examples and verification steps for isomorphism.']}, {'end': 25818.398, 'start': 24780.876, 'title': 'Understanding isomorphisms in mathematics', 'summary': 'Discusses the concept of isomorphisms in mathematics, emphasizing the importance of mental modeling and providing step-by-step examples to demonstrate one-to-one mapping, onto mapping, preservation of addition, and preservation of scalar multiplication.', 'duration': 1037.522, 'highlights': ['Explaining the concept of isomorphisms and importance of mental modeling', 'Demonstrating a step-by-step example of one-to-one mapping', 'Explaining the process of onto mapping using concrete examples and linear systems', 'Detailing the process of preservation of addition and scalar multiplication']}], 'duration': 2412.738, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI23405660.jpg', 'highlights': ['The definition of isomorphism between two vector spaces is a function that is one-to-one, onto, and preserves the structure of the space, with a detailed explanation of verifying these properties.', 'The chapter covers the techniques to show a function is one-to-one, onto, preserves addition, and preserves scalar multiplication.', 'The chapter emphasizes the importance of understanding the steps and the concept of preservation when demonstrating the preservation of addition and scalar multiplication.', 'Explaining the concept of isomorphisms and importance of mental modeling']}, {'end': 28213.014, 'segs': [{'end': 25866.215, 'src': 'embed', 'start': 25837.332, 'weight': 1, 'content': [{'end': 25842.813, 'text': 'the technique that works the most there is the technique that always works with abstraction.', 'start': 25837.332, 'duration': 5.481}, {'end': 25846.314, 'text': 'Anytime you have trouble with abstraction, see if you can make it concrete.', 'start': 25842.913, 'duration': 3.401}, {'end': 25855.747, 'text': 'If you can somehow turn it into numbers instead of letters, that often seems to help a person sort of make sense of it, put it together in their head.', 'start': 25846.781, 'duration': 8.966}, {'end': 25866.215, 'text': "What part is what? Okay, so we're ready for talking about the second half of isomorphisms, and it's really sort of more of the same.", 'start': 25855.867, 'duration': 10.348}], 'summary': 'Technique for working with abstraction: turning it into numbers can help make sense of it.', 'duration': 28.883, 'max_score': 25837.332, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI25837332.jpg'}, {'end': 26001.204, 'src': 'embed', 'start': 25977.399, 'weight': 4, 'content': [{'end': 25984.025, 'text': "We might want to relate a space to itself, so if we relate a space to itself, that's called an automorphism.", 'start': 25977.399, 'duration': 6.626}, {'end': 25987.068, 'text': 'Morphism means map, auto means self, so self-map.', 'start': 25984.205, 'duration': 2.863}, {'end': 25997.203, 'text': "Just as an example, I'm going to take the plane, regular x-y plane, and a dilation map multiplies all vectors by some non-Zehler scalar.", 'start': 25988.549, 'duration': 8.654}, {'end': 25999.744, 'text': 'In my drawing here, I multiplied them all by 1.5.', 'start': 25997.563, 'duration': 2.181}, {'end': 26001.204, 'text': "I couldn't multiply them.", 'start': 25999.744, 'duration': 1.46}], 'summary': 'An automorphism relates a space to itself, demonstrated with a dilation map on the x-y plane.', 'duration': 23.805, 'max_score': 25977.399, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI25977399.jpg'}, {'end': 26495.895, 'src': 'embed', 'start': 26469.142, 'weight': 3, 'content': [{'end': 26473.725, 'text': "but describe something about the spaces that you've probably already intuited in the second chapter.", 'start': 26469.142, 'duration': 4.583}, {'end': 26476.186, 'text': 'You said to yourself, well, this is really just like.', 'start': 26474.205, 'duration': 1.981}, {'end': 26487.987, 'text': 'Yes So, for example, the line y equals 2x is just like the regular real number line in that, for example, 5 times 1, 2 is just like 5.', 'start': 26476.726, 'duration': 11.261}, {'end': 26490.569, 'text': 'And 12 times 1, 2 is just like 12.', 'start': 26487.987, 'duration': 2.582}, {'end': 26491.27, 'text': "Yes, that's right.", 'start': 26490.569, 'duration': 0.701}, {'end': 26495.895, 'text': 'The definition of isomorphism simply makes all that mathematically precise.', 'start': 26492.051, 'duration': 3.844}], 'summary': 'The concept of isomorphism makes mathematical relations precise in chapter two.', 'duration': 26.753, 'max_score': 26469.142, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI26469142.jpg'}, {'end': 26956.081, 'src': 'embed', 'start': 26927.009, 'weight': 2, 'content': [{'end': 26930.151, 'text': 'And the answer will be if, and only if, they have the same dimension.', 'start': 26927.009, 'duration': 3.142}, {'end': 26931.733, 'text': "So that's our big result for the day.", 'start': 26930.212, 'duration': 1.521}, {'end': 26933.915, 'text': 'The idea will be just this simple.', 'start': 26932.233, 'duration': 1.682}, {'end': 26939.551, 'text': 'If you have two vector spaces and you find a basis for the first, maybe it has three vectors.', 'start': 26934.495, 'duration': 5.056}, {'end': 26941.933, 'text': 'You find a basis for the second, maybe it has three vectors.', 'start': 26939.571, 'duration': 2.362}, {'end': 26944.154, 'text': 'Those will be isomorphic vector spaces.', 'start': 26942.273, 'duration': 1.881}, {'end': 26946.796, 'text': 'They are in some sense just like each other.', 'start': 26944.194, 'duration': 2.602}, {'end': 26956.081, 'text': 'And in fact, the isomorphism can be as simple as you take the two basis and associate them.', 'start': 26947.316, 'duration': 8.765}], 'summary': 'Isomorphic vector spaces have same dimension; basis association creates isomorphism.', 'duration': 29.072, 'max_score': 26927.009, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI26927009.jpg'}, {'end': 27665.551, 'src': 'embed', 'start': 27638.121, 'weight': 0, 'content': [{'end': 27641.204, 'text': 'n Every basis has the same number.', 'start': 27638.121, 'duration': 3.083}, {'end': 27644.206, 'text': 'Anyway, they all have n.', 'start': 27642.004, 'duration': 2.202}, {'end': 27652.212, 'text': 'This map that I just illustrated in the previous slide shows you how to associate members of the vector space you gave me with Rn.', 'start': 27644.206, 'duration': 8.006}, {'end': 27659.798, 'text': 'So every finite dimensional vector space, namely every n-dimensional vector space, is isomorphic to Rn.', 'start': 27652.813, 'duration': 6.985}, {'end': 27665.551, 'text': 'So, when I had that picture earlier, where I broke the universe of all vectors,', 'start': 27661.448, 'duration': 4.103}], 'summary': 'Every finite dimensional vector space is isomorphic to rn.', 'duration': 27.43, 'max_score': 27638.121, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI27638121.jpg'}], 'start': 25818.418, 'title': 'Isomorphisms in vector spaces', 'summary': 'Discusses isomorphisms, preservation, and dimension in vector spaces, emphasizing the technique of using abstraction, isomorphisms and automorphisms, preservation of addition and scalar multiplication, and the relationship between vector spaces and rn.', 'chapters': [{'end': 25955.975, 'start': 25818.418, 'title': 'Isomorphisms and preservation', 'summary': 'Discusses the technique of using abstraction, the concept of isomorphisms, and the preservation of addition and scalar multiplication in functions, emphasizing the importance of examples in understanding these concepts.', 'duration': 137.557, 'highlights': ['The technique of using abstraction is suggested to understand difficult concepts, by turning abstract ideas into concrete representations using numbers instead of letters.', 'The chapter discusses isomorphisms and their applications, with an emphasis on the importance of practice and understanding through examples.', 'The preservation of addition property in functions is highlighted, with an example of a map that does not preserve this property, demonstrating the special nature of preservation in functions.']}, {'end': 26332.007, 'start': 25956.776, 'title': 'Isomorphism and automorphism in spaces', 'summary': 'Discusses isomorphisms and automorphisms, emphasizing the concept of self-mapping (automorphism) within a space and its implications, including examples such as dilation, rotation, and reflection, as well as important results about the preservation of scalar multiplication and linear combinations.', 'duration': 375.231, 'highlights': ['Automorphism defined as self-mapping within a space, illustrated with examples like dilation, rotation, and reflection.', 'Implications of automorphisms in revealing uniformity and structure of spaces through rotation and dilation.', 'Results about preservation of scalar multiplication and linear combinations in isomorphisms.']}, {'end': 26900.256, 'start': 26332.047, 'title': 'Isomorphism and preservation in vector spaces', 'summary': 'Introduces isomorphism in vector spaces, showcasing how the line y equals 2x is isomorphic to the real line and demonstrating preservation of addition and scalar multiplication through examples and geometric explanations.', 'duration': 568.209, 'highlights': ['The concept of isomorphism is used to describe the similarity between the line y equals 2x and the real line, emphasizing the fixed nature of 1, 2 and the varying nature of T, providing mathematical precision to previously intuited concepts in the second chapter.', 'Detailed examples and explanations are provided to verify the one-to-one and onto properties, including demonstrating that 7 times 1, 2 maps to 7, showcasing routine verification and preservation of addition and scalar multiplication.', 'Geometric illustrations are used to explain preservation of addition through automorphisms, such as dilation, rotation, and reflection, highlighting how the operations yield the same result whether applied before or after each other.']}, {'end': 27315.261, 'start': 26900.676, 'title': 'Isomorphism and dimension', 'summary': 'Discusses isomorphism between vector spaces, stating that two vector spaces are isomorphic if and only if they have the same dimension, and it illustrates the concept through examples and technical results.', 'duration': 414.585, 'highlights': ['The chapter emphasizes that two vector spaces are isomorphic if and only if they have the same dimension, illustrating it through the association of bases and the preservation of linear combinations by isomorphisms.', 'It explains the concept of isomorphism as a correspondence, a one-to-one and onto map, and highlights that an isomorphism has an inverse function that is also a correspondence.', 'The transcript includes an example of isomorphism between the line through the origin y equals 2x and the real line R1, demonstrating a specific isomorphism and its inverse.', 'It concludes by characterizing the classes of vector spaces based on isomorphism, stating that vector spaces are isomorphic if and only if they have the same dimension.', "The chapter breaks down the characterization of classes of vector spaces into two separate statements: if spaces are isomorphic, then they have the same dimension, and if spaces have the same dimension, then they're isomorphic."]}, {'end': 28213.014, 'start': 27315.661, 'title': 'Vector space isomorphism', 'summary': 'Discusses the concept of vector space isomorphism and its implications, such as the association of vectors in a vector space with their representation in rn, and the characterization that every finite dimensional vector space is isomorphic to rn.', 'duration': 897.353, 'highlights': ['The chapter discusses the concept of vector space isomorphism and its implications, such as the association of vectors in a vector space with their representation in Rn.', 'Every finite dimensional vector space is isomorphic to Rn, as characterized by the representation map, which associates members of a vector space with Rn.', 'Vector space isomorphism provides a complete characterization of when vector spaces are alike, based on their dimensions.']}], 'duration': 2394.596, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI25818418.jpg', 'highlights': ['Every finite dimensional vector space is isomorphic to Rn, characterized by the representation map.', 'The technique of using abstraction is suggested to understand difficult concepts, by turning abstract ideas into concrete representations using numbers instead of letters.', 'The chapter emphasizes that two vector spaces are isomorphic if and only if they have the same dimension, illustrating it through the association of bases and the preservation of linear combinations by isomorphisms.', 'The concept of isomorphism is used to describe the similarity between the line y equals 2x and the real line, emphasizing the fixed nature of 1, 2 and the varying nature of T, providing mathematical precision to previously intuited concepts in the second chapter.', 'Automorphism defined as self-mapping within a space, illustrated with examples like dilation, rotation, and reflection.']}, {'end': 29482.129, 'segs': [{'end': 28275.964, 'src': 'embed', 'start': 28233.105, 'weight': 1, 'content': [{'end': 28236.546, 'text': "Well, of course what we're doing for real is we're really working with polynomials,", 'start': 28233.105, 'duration': 3.441}, {'end': 28240.027, 'text': 'but we can represent it in some way by working with three tall vectors.', 'start': 28236.546, 'duration': 3.481}, {'end': 28241.963, 'text': 'Okay, very good.', 'start': 28241.283, 'duration': 0.68}, {'end': 28248.167, 'text': 'Now, last time we talked about the fact that we are going to start something new.', 'start': 28242.264, 'duration': 5.903}, {'end': 28250.408, 'text': 'This is a new second section of the third chapter.', 'start': 28248.187, 'duration': 2.221}, {'end': 28254.951, 'text': 'So the new thing is, in some sense, easier than the old thing.', 'start': 28250.789, 'duration': 4.162}, {'end': 28258.833, 'text': 'Isomorphism has two different properties.', 'start': 28256.132, 'duration': 2.701}, {'end': 28260.274, 'text': "First of all, it's a correspondence.", 'start': 28258.913, 'duration': 1.361}, {'end': 28263.316, 'text': 'Second of all, it preserves linear combinations.', 'start': 28260.614, 'duration': 2.702}, {'end': 28269.98, 'text': "Homomorphisms will have only one of those two properties, and so in that sense they're easier than isomorphisms.", 'start': 28264.397, 'duration': 5.583}, {'end': 28275.964, 'text': 'However, homomorphisms are less conceptually clear than isomorphisms.', 'start': 28271.522, 'duration': 4.442}], 'summary': 'Working with polynomials using tall vectors, easier new section, isomorphisms vs homomorphisms', 'duration': 42.859, 'max_score': 28233.105, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI28233105.jpg'}, {'end': 28372.042, 'src': 'embed', 'start': 28350.85, 'weight': 2, 'content': [{'end': 28361.056, 'text': "the definition of a homomorphism is a function between vector spaces that preserves addition and of course we've seen the preserves addition definition before and preserves scalar multiplication,", 'start': 28350.85, 'duration': 10.206}, {'end': 28362.177, 'text': "and again we've seen that before.", 'start': 28361.056, 'duration': 1.121}, {'end': 28364.698, 'text': "It's called a homomorphism or a linear map.", 'start': 28362.497, 'duration': 2.201}, {'end': 28372.042, 'text': "Those two terms are both very widespread and I'll try to, maybe not exactly strictly speaking, alternate from one to the other,", 'start': 28365.038, 'duration': 7.004}], 'summary': 'Homomorphism: function preserving addition and scalar multiplication', 'duration': 21.192, 'max_score': 28350.85, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI28350850.jpg'}, {'end': 28661.538, 'src': 'embed', 'start': 28631.638, 'weight': 4, 'content': [{'end': 28632.198, 'text': "It doesn't have that.", 'start': 28631.638, 'duration': 0.56}, {'end': 28633.518, 'text': "So it's a Greek letter iota.", 'start': 28632.218, 'duration': 1.3}, {'end': 28641.84, 'text': 'Anyway, the inclusion map from R2 to R3 takes two tall vectors and makes three tall vectors, outputs three tall vectors.', 'start': 28633.918, 'duration': 7.922}, {'end': 28643.64, 'text': "And that's a homomorphism.", 'start': 28642.3, 'duration': 1.34}, {'end': 28651.668, 'text': "The verification, we've done the verification for isomorphisms an awful lot, so this is just not going to be a surprise to anybody.", 'start': 28645.962, 'duration': 5.706}, {'end': 28657.995, 'text': 'If you apply iota to a linear combination, then you get the linear combination of the iotas.', 'start': 28652.148, 'duration': 5.847}, {'end': 28661.538, 'text': "I'll go through the steps, but it isn't anything we haven't seen before.", 'start': 28658.876, 'duration': 2.662}], 'summary': 'Inclusion map from r2 to r3 transforms 2 tall vectors into 3 tall vectors, a homomorphism.', 'duration': 29.9, 'max_score': 28631.638, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI28631638.jpg'}, {'end': 28736.703, 'src': 'embed', 'start': 28708.936, 'weight': 0, 'content': [{'end': 28713.238, 'text': "Now here's one that I think is new and interesting, I think.", 'start': 28708.936, 'duration': 4.302}, {'end': 28721.836, 'text': 'Okay, it says the derivative function, the regular derivative function from calculus 1, is a transformation.', 'start': 28715.66, 'duration': 6.176}, {'end': 28724.038, 'text': "It's a function on polynomial spaces.", 'start': 28722.237, 'duration': 1.801}, {'end': 28726.859, 'text': 'It takes one polynomial space and outputs another polynomial.', 'start': 28724.138, 'duration': 2.721}, {'end': 28733.862, 'text': 'So it takes vectors, for example, takes quadratic polynomials and outputs linear polynomials.', 'start': 28727.159, 'duration': 6.703}, {'end': 28736.703, 'text': "I'm writing the derivative as d dx.", 'start': 28734.702, 'duration': 2.001}], 'summary': 'The derivative function in calculus 1 transforms polynomial spaces, converting quadratic polynomials to linear polynomials.', 'duration': 27.767, 'max_score': 28708.936, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI28708936.jpg'}], 'start': 28213.514, 'title': 'Linear transformations and representations', 'summary': 'Covers the representation of polynomials using three tall vectors, properties of isomorphisms and homomorphisms in vector spaces, derivative function as a transformation on polynomial spaces, and the definition of homomorphism with examples in linear maps.', 'chapters': [{'end': 28254.951, 'start': 28213.514, 'title': 'Representation with three tall vectors', 'summary': 'Discusses the representation of polynomials using three tall vectors, introducing a new, easier concept in the second section of the third chapter.', 'duration': 41.437, 'highlights': ['The chapter discusses representing polynomials with three tall vectors, facilitating easier understanding and manipulation of the data.', 'The new concept introduced in the second section of the third chapter is described as easier than the previous concept.']}, {'end': 28707.035, 'start': 28256.132, 'title': 'Isomorphisms & homomorphisms in vector spaces', 'summary': 'Discusses the properties of isomorphisms and homomorphisms in vector spaces, emphasizing the differences between them and providing examples to illustrate their concepts and applications.', 'duration': 450.903, 'highlights': ['Isomorphisms are conceptually clearer than homomorphisms and preserve both correspondence and linear combinations, making them more complex than homomorphisms.', 'Homomorphisms are defined as functions between vector spaces that preserve addition and scalar multiplication, and every isomorphism is automatically a homomorphism with extra special conditions.', 'The inclusion map from R2 to R3 is a homomorphism that outputs three tall vectors and preserves linear combinations, as verified through routine steps.', 'The zero map is a trivial but true homomorphism that takes any input and maps it to W, preserving linear combinations.', 'The chapter also emphasizes the differences between isomorphisms and homomorphisms, highlighting their respective complexities and applications in vector spaces.']}, {'end': 28998.468, 'start': 28708.936, 'title': 'Derivative and trace functions in linear maps', 'summary': 'Discusses the derivative function as a transformation on polynomial spaces, demonstrating its application to quadratic polynomials to output linear polynomials, and explores the trace map as a function on two by two matrices to output real numbers, emphasizing the familiarity of these linear maps and introducing unfamiliar examples.', 'duration': 289.532, 'highlights': ['The derivative function is presented as a transformation on polynomial spaces, demonstrating its application to quadratic polynomials to output linear polynomials, highlighting the familiar formula ddx of ax squared plus bx plus c as 2ax plus b and emphasizing its relevance to calculus 1.', 'The trace map is introduced as a function on two by two matrices to output real numbers, with a focus on its application to linear combinations and the emphasis on the familiarity of these linear maps, contrasting familiar and unfamiliar examples.']}, {'end': 29482.129, 'start': 28998.808, 'title': 'Homomorphism and linear maps', 'summary': 'Discusses the definition of homomorphism and provides examples, including the mechanics of linear maps and the process of checking for homomorphisms in various scenarios.', 'duration': 483.321, 'highlights': ['The process of checking for homomorphisms in various scenarios', 'The definition of homomorphism and its examples', 'Detailed examination of linear maps']}], 'duration': 1268.615, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI28213514.jpg', 'highlights': ['The derivative function transforms polynomial spaces, demonstrating its relevance to calculus 1.', 'Isomorphisms preserve correspondence and linear combinations, making them more complex than homomorphisms.', 'Homomorphisms are functions between vector spaces that preserve addition and scalar multiplication.', 'The chapter discusses representing polynomials with three tall vectors, facilitating easier understanding and manipulation of the data.', 'The inclusion map from R2 to R3 is a homomorphism that outputs three tall vectors and preserves linear combinations.']}, {'end': 31729.626, 'segs': [{'end': 29591.784, 'src': 'embed', 'start': 29563.992, 'weight': 5, 'content': [{'end': 29573.871, 'text': 'okay, just all the stuff that we talked about last time, and it brings us to the theorem that I want to cover today.', 'start': 29563.992, 'duration': 9.879}, {'end': 29578.136, 'text': 'The theorem says that a homomorphism is determined by its action on a basis.', 'start': 29573.891, 'duration': 4.245}, {'end': 29581.621, 'text': 'If you tell me what the basis is and you say what?', 'start': 29578.777, 'duration': 2.844}, {'end': 29586.86, 'text': 'what does the first basis element map to?', 'start': 29583.737, 'duration': 3.123}, {'end': 29589.222, 'text': 'what does the second basis element map to?', 'start': 29586.86, 'duration': 2.362}, {'end': 29591.784, 'text': 'what does the third basis element map to, etc?', 'start': 29589.222, 'duration': 2.562}], 'summary': 'Homomorphism is determined by its action on a basis.', 'duration': 27.792, 'max_score': 29563.992, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI29563992.jpg'}, {'end': 29986.183, 'src': 'embed', 'start': 29960.427, 'weight': 4, 'content': [{'end': 29969.112, 'text': 'the action of the map on the basis, and I extend linearly to the entire space to get the action of the map on any element of the space.', 'start': 29960.427, 'duration': 8.685}, {'end': 29973.295, 'text': 'You figure out what are the coefficients, and you put those coefficients down.', 'start': 29969.693, 'duration': 3.602}, {'end': 29981.32, 'text': "Here's an example to try to illustrate just how useful this is.", 'start': 29975.957, 'duration': 5.363}, {'end': 29983.301, 'text': "So here's an example.", 'start': 29982.281, 'duration': 1.02}, {'end': 29986.183, 'text': "I'm going to figure out the formula for rotating in the plane.", 'start': 29983.401, 'duration': 2.782}], 'summary': 'Linearly extending map action to whole space, determining coefficients, and illustrating with rotating in plane formula.', 'duration': 25.756, 'max_score': 29960.427, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI29960427.jpg'}, {'end': 30388.333, 'src': 'embed', 'start': 30365.29, 'weight': 1, 'content': [{'end': 30374.279, 'text': 'And I want to finish today by observing something about linear transformations, about linear functions from V to W.', 'start': 30365.29, 'duration': 8.989}, {'end': 30380.125, 'text': "So if you look at the vector spaces V and W, then the set of linear functions from V to W, that's a set.", 'start': 30374.279, 'duration': 5.846}, {'end': 30383.808, 'text': 'You have a whole bunch of different linear functions from V to W.', 'start': 30380.225, 'duration': 3.583}, {'end': 30385.59, 'text': "I'm observing something about that.", 'start': 30383.808, 'duration': 1.782}, {'end': 30388.333, 'text': 'That set is in fact itself a vector space.', 'start': 30385.93, 'duration': 2.403}], 'summary': 'Linear functions from v to w form a vector space.', 'duration': 23.043, 'max_score': 30365.29, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI30365290.jpg'}, {'end': 31238.746, 'src': 'embed', 'start': 31208.337, 'weight': 0, 'content': [{'end': 31210.562, 'text': 'The range space, then, is the image of the entire space.', 'start': 31208.337, 'duration': 2.225}, {'end': 31217.496, 'text': "So it's the special case of the previous result that takes the subspace to be the entire space.", 'start': 31211.312, 'duration': 6.184}, {'end': 31219.777, 'text': 'It takes the subspace of V to be the entire space.', 'start': 31217.516, 'duration': 2.261}, {'end': 31222.079, 'text': 'I write it with a script R there.', 'start': 31220.398, 'duration': 1.681}, {'end': 31224.38, 'text': "There's different notations you see around.", 'start': 31222.139, 'duration': 2.241}, {'end': 31228.082, 'text': 'If you Google around, you might see some people write R-A-N for range.', 'start': 31224.42, 'duration': 3.662}, {'end': 31229.683, 'text': "You'll see other things also.", 'start': 31228.422, 'duration': 1.261}, {'end': 31238.746, 'text': 'I think script R is a compromise between writing it relatively short and being relatively clear.', 'start': 31230.063, 'duration': 8.683}], 'summary': 'The range space is the entire space, denoted as script r, a compromise between short and clear.', 'duration': 30.409, 'max_score': 31208.337, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI31208337.jpg'}, {'end': 31399.164, 'src': 'embed', 'start': 31374.099, 'weight': 2, 'content': [{'end': 31379.1, 'text': 'So the rank of the derivative function is the dimension of the cubic polynomials.', 'start': 31374.099, 'duration': 5.001}, {'end': 31382.941, 'text': "and of course the cubic polynomials don't be misled by the subscript 3..", 'start': 31379.1, 'duration': 3.841}, {'end': 31389.142, 'text': 'The cubic polynomials are not dimension 3, the cubic polynomials are dimension 4, because you have constants coefficients of x,', 'start': 31382.941, 'duration': 6.201}, {'end': 31391.502, 'text': 'coefficients of x squared and coefficients of x cubed.', 'start': 31389.142, 'duration': 2.36}, {'end': 31392.843, 'text': "so there's sort of four degrees of freedom.", 'start': 31391.502, 'duration': 1.341}, {'end': 31399.164, 'text': "Okay, another example, we're going to talk a lot about the geometry here today.", 'start': 31392.863, 'duration': 6.301}], 'summary': 'The rank of the derivative function is the dimension of the cubic polynomials, which have 4 degrees of freedom.', 'duration': 25.065, 'max_score': 31374.099, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI31374099.jpg'}, {'end': 31455.978, 'src': 'embed', 'start': 31429.96, 'weight': 3, 'content': [{'end': 31437.585, 'text': 'So you can imagine if you stand on the point 5, 6 in the plane and look up, you see all those points above you that project down.', 'start': 31429.96, 'duration': 7.625}, {'end': 31440.047, 'text': 'Or look down, you see all the points below you that project up.', 'start': 31437.786, 'duration': 2.261}, {'end': 31455.978, 'text': 'So the rank of the projection map is 2 because the output space, the range space, in this case the range of the entire map, is 2-dimensional.', 'start': 31443.369, 'duration': 12.609}], 'summary': 'The projection map has a rank of 2, as it has a 2-dimensional range space.', 'duration': 26.018, 'max_score': 31429.96, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI31429960.jpg'}], 'start': 29484.151, 'title': 'Homomorphism and linear maps', 'summary': 'Discusses the homomorphism theorem, linear maps, linear transformations, linear functions, subspaces, range spaces, and projection maps, with examples and emphasis on rank, range space, and dimensional properties.', 'chapters': [{'end': 29635.765, 'start': 29484.151, 'title': 'Homomorphism theorem and linear map', 'summary': 'Discusses the theorem that a homomorphism is determined by its action on a basis, stating that there is one and only one homomorphism with a given action, and it is unique, as well as the definition and properties of homomorphism and linear map.', 'duration': 151.614, 'highlights': ['The theorem states that a homomorphism is determined by its action on a basis, with one and only one homomorphism with a given action and it is unique.', 'The definition of homomorphism and linear map involves preserving addition and scalar multiplication, and the theorem emphasizes that a homomorphism is determined by its action on a basis.']}, {'end': 30066.954, 'start': 29636.716, 'title': 'Linear maps and homomorphism', 'summary': 'Discusses the concept of linear maps and homomorphism, illustrating how a map is determined by its action on a basis and extending linearly to the entire space, with examples including rotating vectors in the plane.', 'duration': 430.238, 'highlights': ['A linear map is determined by its action on a basis, allowing the computation of its action on any other member of the domain, as demonstrated by finding the action of H on a given vector.', 'Extending linearly allows a function defined on a basis to be extended to a function defined on all of v, providing the ability to figure out the action of a map on any element of the space.', 'The process of rotating vectors in the plane through an angle counterclockwise of theta is demonstrated, with the formula for the geometric action obtained by fixing a basis for the domain and determining where the basis vectors map under the desired action.']}, {'end': 30340.237, 'start': 30068.015, 'title': 'Linear transformation and homomorphism', 'summary': 'Discusses linear transformations and homomorphisms, illustrating with examples such as rotation formulas, extending linearly for quadratic polynomials, and reflection across the yz plane.', 'duration': 272.222, 'highlights': ["The formula for rotation of x, y's in the plane is obtained by specifying where the basis vectors go, followed by working through the details of homomorphism, illustrating the concept with examples.", 'Defining a map eval sub 3 by specifying where it sends members of the quadratic polynomials basis, showing the linearity of the map by extending it to any quadratic polynomial, and providing an example of evaluating the map on a specific polynomial.', 'Explaining the concept of linear transformation as a synonym for a linear map and homomorphism, and the preference to use transformation specifically when the domain and co-domain are the same, with a brief note on terminology differences in other sources.', 'Illustrating the reflection of vectors over the YZ plane as an example of a linear transformation, providing specific examples of vectors and their reflections.']}, {'end': 31126.424, 'start': 30340.239, 'title': 'Linear functions and vector spaces', 'summary': 'Discusses linear transformations, linear functions from v to w, and the set of linear functions from v to w being a vector space, and extends to discussing range space and null space, demonstrating their properties through examples.', 'duration': 786.185, 'highlights': ['The set of linear functions from V to W is itself a vector space', 'Linear maps from R to R2 is a dimension two vector space', 'Illustration of homomorphism properties through examples', 'Demonstration of range space and null space properties through examples']}, {'end': 31392.843, 'start': 31128.871, 'title': 'Subspaces and range spaces', 'summary': 'Discusses subspaces and range spaces in 2x2 matrices, illustrating how they form a two-dimensional subspace of the four-dimensional space and using examples such as rotation and derivative maps to explain the concept, with an emphasis on the rank of the derivative function and the dimension of the cubic polynomials.', 'duration': 263.972, 'highlights': ['The chapter discusses subspaces and range spaces in 2x2 matrices, illustrating how they form a two-dimensional subspace of the four-dimensional space.', 'The function that rotates vectors counterclockwise to the angle theta is a linear map, and each line through the origin in the domain is a subspace.', "The range space, denoted as script R, is the image of the entire space, and the dimension of the range space is called the map's rank.", 'The rank of the derivative function is the dimension of the cubic polynomials, which are not dimension 3 but dimension 4 due to the coefficients of x, x squared, and x cubed.']}, {'end': 31729.626, 'start': 31392.863, 'title': 'Projection map and homomorphism', 'summary': 'Discusses the projection map from r3 to r2, highlighting its rank, range space, and the significance of dropping the one-to-one condition in homomorphism, emphasizing the need to form a mental conception of homomorphisms.', 'duration': 336.763, 'highlights': ['The range space of the projection map from R3 to R2 is 2-dimensional, with infinitely many vectors projecting to a point, illustrating the rank and significance of the range space in this context.', 'Dropping the one-to-one condition in homomorphism allows an output vector to have many associated inputs, leading to the concept of inverse image sets and the need to understand the structure of these sets for building a mental conception of homomorphisms.', 'Discussion on the preservation of addition condition in homomorphism and the importance of forming a mental model of homomorphisms to understand their purpose and function.', 'Illustration of the projection map from R2 to R, demonstrating the elements of inverse image sets and the preservation of addition condition, emphasizing the need to comprehend the practical implications of homomorphisms.']}], 'duration': 2245.475, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI29484151.jpg', 'highlights': ["The range space, denoted as script R, is the image of the entire space, and the dimension of the range space is called the map's rank.", 'The set of linear functions from V to W is itself a vector space', 'The rank of the derivative function is the dimension of the cubic polynomials, which are not dimension 3 but dimension 4 due to the coefficients of x, x squared, and x cubed.', 'The range space of the projection map from R3 to R2 is 2-dimensional, with infinitely many vectors projecting to a point, illustrating the rank and significance of the range space in this context.', 'A linear map is determined by its action on a basis, allowing the computation of its action on any other member of the domain, as demonstrated by finding the action of H on a given vector.', 'The theorem states that a homomorphism is determined by its action on a basis, with one and only one homomorphism with a given action and it is unique.']}, {'end': 33774.354, 'segs': [{'end': 32081.786, 'src': 'embed', 'start': 32053.021, 'weight': 1, 'content': [{'end': 32057.462, 'text': 'The intuition is that linear maps organize the domain into inverse images.', 'start': 32053.021, 'duration': 4.441}, {'end': 32063.257, 'text': 'in such a way that the sets reflect the structure of the range, red plus blue makes magenta.', 'start': 32058.575, 'duration': 4.682}, {'end': 32066.839, 'text': 'So what do I mean by organize the domain into inverse images?', 'start': 32063.958, 'duration': 2.881}, {'end': 32073.802, 'text': 'Well, remember when we looked at projection from R3 to R2, and you stood on 5, 6, and you looked up and looked down.', 'start': 32066.979, 'duration': 6.823}, {'end': 32076.844, 'text': 'those are all the vectors associated with your current location.', 'start': 32073.802, 'duration': 3.042}, {'end': 32081.786, 'text': 'If you paint your current location red, all of those vectors would become red.', 'start': 32077.324, 'duration': 4.462}], 'summary': "Linear maps organize domain into inverse images, reflecting range's structure.", 'duration': 28.765, 'max_score': 32053.021, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI32053021.jpg'}, {'end': 32409.97, 'src': 'embed', 'start': 32384.821, 'weight': 0, 'content': [{'end': 32390.143, 'text': "When we come back next time, we're going to understand what those inverse images are.", 'start': 32384.821, 'duration': 5.322}, {'end': 32395.546, 'text': "It's related to what's called the null space, the inverse image of the origin.", 'start': 32390.804, 'duration': 4.742}, {'end': 32401.069, 'text': 'Okay, so we started off by talking about the range space today.', 'start': 32396.887, 'duration': 4.182}, {'end': 32409.97, 'text': 'Homomorphisms organize the range space into these inverse images in such a way that the inverse images have a structure.', 'start': 32401.601, 'duration': 8.369}], 'summary': 'Homomorphisms organize range space into inverse images, related to null space.', 'duration': 25.149, 'max_score': 32384.821, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI32384821.jpg'}, {'end': 32479.691, 'src': 'embed', 'start': 32438.2, 'weight': 4, 'content': [{'end': 32444.263, 'text': 'Okay, so H is going to take the real three space over to real one space to the line.', 'start': 32438.2, 'duration': 6.063}, {'end': 32451.159, 'text': "And it's going to send x, y, z to x minus y minus 2z.", 'start': 32444.743, 'duration': 6.416}, {'end': 32459.627, 'text': "OK, and I'm asked to find first h inverse of 1.", 'start': 32456.664, 'duration': 2.963}, {'end': 32466.173, 'text': "Of course, 1 is a member of the range, so that's a perfectly sensible question.", 'start': 32459.627, 'duration': 6.546}, {'end': 32479.691, 'text': "So this is the set of xyz's that map to 1.", 'start': 32467.635, 'duration': 12.056}], 'summary': 'H maps 3d space to 1d space by x-y-2z. finding h inverse of 1.', 'duration': 41.491, 'max_score': 32438.2, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI32438200.jpg'}, {'end': 32895.536, 'src': 'embed', 'start': 32860.833, 'weight': 9, 'content': [{'end': 32863.195, 'text': "So I'm going to look at a couple of examples.", 'start': 32860.833, 'duration': 2.362}, {'end': 32864.636, 'text': 'You see a couple of examples on this slide.', 'start': 32863.214, 'duration': 1.422}, {'end': 32866.917, 'text': 'The derivative map from P2 to P1.', 'start': 32865.116, 'duration': 1.801}, {'end': 32873.302, 'text': 'The null space is all of the quadratic polynomials whose derivative is zero.', 'start': 32868.138, 'duration': 5.164}, {'end': 32876.604, 'text': '2AX plus B makes zero.', 'start': 32874.603, 'duration': 2.001}, {'end': 32886.949, 'text': 'Now, remember that 0 is the 0 polynomial, 0x plus 0, element of P1, 0x plus 0.', 'start': 32878.584, 'duration': 8.365}, {'end': 32895.536, 'text': 'So 2ax plus b equals 0 means that they have the same constant coefficient, namely b is 0, and the same x coefficient, so 2a equals 0.', 'start': 32886.949, 'duration': 8.587}], 'summary': 'The transcript explains the null space of a derivative map from p2 to p1, using examples and equations.', 'duration': 34.703, 'max_score': 32860.833, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI32860833.jpg'}, {'end': 32941.303, 'src': 'embed', 'start': 32915.739, 'weight': 7, 'content': [{'end': 32922.784, 'text': "And of course that's the familiar result from calculus 1, that if you want to see which functions have a derivative of 0,", 'start': 32915.739, 'duration': 7.045}, {'end': 32926.407, 'text': "which polynomial functions have a derivative of 0,, it's all the ones with a constant term.", 'start': 32922.784, 'duration': 3.623}, {'end': 32930.389, 'text': 'Constant term could be anything, and the other part disappear.', 'start': 32926.947, 'duration': 3.442}, {'end': 32936.012, 'text': 'So the null space is all the constant polynomials.', 'start': 32932.91, 'duration': 3.102}, {'end': 32941.303, 'text': 'Nullity is 1.', 'start': 32938.021, 'duration': 3.282}], 'summary': 'In calculus 1, functions with derivative 0 are constant polynomials. nullity is 1.', 'duration': 25.564, 'max_score': 32915.739, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI32915739.jpg'}, {'end': 33000.538, 'src': 'embed', 'start': 32969.249, 'weight': 6, 'content': [{'end': 32971.309, 'text': "So it's clearly a dimension one space.", 'start': 32969.249, 'duration': 2.06}, {'end': 32976.211, 'text': "It's all multiples of that one size one basis.", 'start': 32972.65, 'duration': 3.561}, {'end': 32978.512, 'text': 'And so the null it is one.', 'start': 32977.43, 'duration': 1.082}, {'end': 32982.853, 'text': 'More examples of finding the null space.', 'start': 32980.852, 'duration': 2.001}, {'end': 32987.814, 'text': "Here's a homomorphism from the two by two matrices to the two tall vectors.", 'start': 32983.292, 'duration': 4.522}, {'end': 32992.316, 'text': 'A, B, C, D maps over to A plus B, C plus D.', 'start': 32988.315, 'duration': 4.001}, {'end': 33000.538, 'text': 'For the null space, you get that A plus B has to make zero, and C plus D has to make zero.', 'start': 32994.335, 'duration': 6.203}], 'summary': 'Discussing null space in a one-dimensional space and homomorphism from 2x2 matrices to 2-tall vectors.', 'duration': 31.289, 'max_score': 32969.249, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI32969249.jpg'}, {'end': 33398.8, 'src': 'embed', 'start': 33368.841, 'weight': 2, 'content': [{'end': 33370.661, 'text': "You're standing at the point 5, 6.", 'start': 33368.841, 'duration': 1.82}, {'end': 33371.884, 'text': 'You look up, you look down.', 'start': 33370.663, 'duration': 1.221}, {'end': 33376.027, 'text': "There's a line's worth of vectors that all collapse to be where you're standing.", 'start': 33372.023, 'duration': 4.004}, {'end': 33380.83, 'text': 'Three-dimensional domain collapses to be the two-dimensional range.', 'start': 33377.828, 'duration': 3.002}, {'end': 33382.231, 'text': 'Where did that extra dimension go??', 'start': 33380.87, 'duration': 1.361}, {'end': 33385.693, 'text': 'And the answer is all of the lines, all the vectors above and below.', 'start': 33382.491, 'duration': 3.202}, {'end': 33388.735, 'text': "that entire line's worth of vectors collapses to that one point.", 'start': 33385.693, 'duration': 3.042}, {'end': 33391.055, 'text': 'So the nullity is 1.', 'start': 33389.915, 'duration': 1.14}, {'end': 33398.8, 'text': 'In particular, they vectors above and below the origin, collapsed to be that one point.', 'start': 33391.055, 'duration': 7.745}], 'summary': 'In a 3d space, a line of vectors collapses to a single point, resulting in a nullity of 1.', 'duration': 29.959, 'max_score': 33368.841, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI33368841.jpg'}, {'end': 33466.737, 'src': 'embed', 'start': 33441.896, 'weight': 3, 'content': [{'end': 33449.263, 'text': 'And so the domain is organized into inverse images that are vertical lines, one-dimensional sets that look like the null space.', 'start': 33441.896, 'duration': 7.367}, {'end': 33452.845, 'text': 'And so you have sort of two dimensions of freedom, and one dimension has gone away.', 'start': 33449.603, 'duration': 3.242}, {'end': 33454.688, 'text': 'Where did it go? Well, project it down to zero.', 'start': 33452.866, 'duration': 1.822}, {'end': 33460.332, 'text': 'Okay, now there are just really a couple of quick examples.', 'start': 33457.97, 'duration': 2.362}, {'end': 33466.737, 'text': 'The derivative function here has this for our range space, and this is its null space.', 'start': 33460.432, 'duration': 6.305}], 'summary': 'The domain is organized into inverse images, with two dimensions of freedom and one dimension projected down to zero.', 'duration': 24.841, 'max_score': 33441.896, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI33441896.jpg'}, {'end': 33521.487, 'src': 'embed', 'start': 33488.386, 'weight': 5, 'content': [{'end': 33489.927, 'text': 'The rank is 2, the nullity is 2.', 'start': 33488.386, 'duration': 1.541}, {'end': 33495.328, 'text': 'So again, the example here is that the null space can have dimension 0.', 'start': 33489.927, 'duration': 5.401}, {'end': 33496.369, 'text': "Don't leave that case out.", 'start': 33495.328, 'duration': 1.041}, {'end': 33507.932, 'text': "Don't forget the case that if the null space is only the trivial subspace of the domain, then the nullity is zero.", 'start': 33496.409, 'duration': 11.523}, {'end': 33511.314, 'text': "We're going to compute more null spaces when we get to the next section.", 'start': 33507.953, 'duration': 3.361}, {'end': 33515.894, 'text': "We're going to compute more range and null spaces when we get to the next section, just as a heads up.", 'start': 33511.334, 'duration': 4.56}, {'end': 33521.487, 'text': 'Just a couple more, just to finish, a couple more results here.', 'start': 33518.446, 'duration': 3.041}], 'summary': 'Rank is 2, nullity is 2. null space dimension can be 0. nullity is zero for trivial null space. more computations in next section.', 'duration': 33.101, 'max_score': 33488.386, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI33488386.jpg'}, {'end': 33689.05, 'src': 'embed', 'start': 33661.276, 'weight': 8, 'content': [{'end': 33668.945, 'text': 'Okay, so we have been working through the definition of homomorphism, and I want to make a couple comments about where we are.', 'start': 33661.276, 'duration': 7.669}, {'end': 33675.152, 'text': "So most folks who take this course don't really struggle too much with the solving of linear systems.", 'start': 33669.666, 'duration': 5.486}, {'end': 33676.493, 'text': 'It seems very concrete to them.', 'start': 33675.172, 'duration': 1.321}, {'end': 33679.056, 'text': 'You follow these steps, you always get the right answer.', 'start': 33676.854, 'duration': 2.202}, {'end': 33680.538, 'text': 'That seems very sensible.', 'start': 33679.677, 'duration': 0.861}, {'end': 33689.05, 'text': 'The next step up in abstraction, not very abstract, the next step up in abstraction is to study collections of linear combinations.', 'start': 33681.223, 'duration': 7.827}], 'summary': 'Homomorphism definition discussed, linear systems solving is concrete for most, next step is studying collections of linear combinations.', 'duration': 27.774, 'max_score': 33661.276, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI33661276.jpg'}], 'start': 31730.707, 'title': 'Homomorphisms and linear maps', 'summary': 'Demonstrates how linear maps and homomorphisms organize the domain and range spaces into inverse images, and explains the concept of null space and the dimensions of domain and range, with examples such as projections and derivative maps.', 'chapters': [{'end': 32261.743, 'start': 31730.707, 'title': 'Linear maps and vector addition', 'summary': 'Demonstrates how linear maps organize the domain into inverse images, with red plus blue vectors making magenta vectors and homomorphisms dividing the output space into components.', 'duration': 531.036, 'highlights': ['Linear maps organize the domain into inverse images', 'Red plus blue vectors make magenta vectors', 'Vectors that add to 2 and 3 combine to add to 5']}, {'end': 32677.283, 'start': 32267.267, 'title': 'Understanding homomorphisms and inverse images', 'summary': 'Explains how homomorphisms organize the range space into inverse images, and demonstrates this with an example of mapping real three space to real one space, showcasing the calculations for h inverse of 1, h inverse of -1, and verifying vector addition for h inverse of 0.', 'duration': 410.016, 'highlights': ['Homomorphisms organize the range space into these inverse images in such a way that the inverse images have a structure.', 'Demonstrates the calculations for h inverse of 1, h inverse of -1, and verifying vector addition for h inverse of 0, showcasing the process of finding inverse images and confirming their properties.', 'Explains the concept of inverse images and their relationship with the null space, highlighting the upcoming focus on understanding the appearance of inverse images.']}, {'end': 33071.348, 'start': 32677.323, 'title': 'Homomorphism and inverse images', 'summary': 'Explores the concept of homomorphism, inverse images, and null space, providing examples and explanations of the nullity for various homomorphisms, including the null space of a derivative map and a 2x2 matrix homomorphism.', 'duration': 394.025, 'highlights': ['The null space is all the constant polynomials. Nullity is 1.', 'The null space is all multiples of the vector (-1, 1). Nullity is 1.', 'The null space is a dimension two subspace, so the nullity is two.', 'The nullity is zero because it has a basis of psi zero.']}, {'end': 33774.354, 'start': 33073.618, 'title': 'Homomorphism and linear transformation', 'summary': 'Discusses the concept of rank and nullity in linear transformations, illustrating how the dimension of the domain collapses to the dimension of the range, with examples of projection from r3 to r2 and the derivative function, emphasizing the importance of understanding homomorphisms and the need for concrete numerical examples to solidify comprehension.', 'duration': 700.736, 'highlights': ['The dimension of the domain collapses to the dimension of the range', 'Examples of projection from R3 to R2 and the derivative function', 'Importance of understanding homomorphisms and the need for concrete numerical examples']}], 'duration': 2043.647, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI31730707.jpg', 'highlights': ['Homomorphisms organize the range space into inverse images', 'Linear maps organize the domain into inverse images', 'The dimension of the domain collapses to the dimension of the range', 'Explains the concept of null space and the dimensions of domain and range', 'Demonstrates the calculations for h inverse of 1, h inverse of -1, and verifying vector addition', 'The null space is a dimension two subspace, so the nullity is two', 'The null space is all multiples of the vector (-1, 1). Nullity is 1', 'The null space is all the constant polynomials. Nullity is 1', 'Importance of understanding homomorphisms and the need for concrete numerical examples', 'Examples of projection from R3 to R2 and the derivative function']}, {'end': 35442.02, 'segs': [{'end': 33874.021, 'src': 'embed', 'start': 33844.055, 'weight': 1, 'content': [{'end': 33858.299, 'text': 'That is to say, these inverse images of points from the range, these ways that the homomorphism organizes the domains, have a structure to them.', 'start': 33844.055, 'duration': 14.244}, {'end': 33862.445, 'text': 'They add according to the rules that the definition of homomorphism gives you.', 'start': 33858.66, 'duration': 3.785}, {'end': 33869.074, 'text': 'And we saw a number of examples and worked through some.', 'start': 33864.047, 'duration': 5.027}, {'end': 33869.815, 'text': 'do it in this case.', 'start': 33869.074, 'duration': 0.741}, {'end': 33870.436, 'text': 'do it in this case.', 'start': 33869.815, 'duration': 0.621}, {'end': 33874.021, 'text': 'do it in this case, so a person gets some feel for how exactly it works.', 'start': 33870.436, 'duration': 3.585}], 'summary': 'Homomorphisms organize domains with defined rules and structures.', 'duration': 29.966, 'max_score': 33844.055, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI33844055.jpg'}, {'end': 33926.566, 'src': 'embed', 'start': 33892.105, 'weight': 0, 'content': [{'end': 33895.466, 'text': "And so I'm going to have some homomorphisms and say what do they do in the plane?", 'start': 33892.105, 'duration': 3.361}, {'end': 33909.078, 'text': 'Okay so, last time we did the sort of algebraic expression of this homomorphism divides the domain up into inverse images of points in the range.', 'start': 33897.492, 'duration': 11.586}, {'end': 33916.621, 'text': 'The algebraic way to state that is to say the nullity plus the rank equals the dimension of the domain.', 'start': 33911.499, 'duration': 5.122}, {'end': 33926.566, 'text': "That if you're looking at the homomorphism of projection and you're standing in the plane at There's an entire dimension's worth,", 'start': 33917.022, 'duration': 9.544}], 'summary': 'Homomorphisms divide domain into inverse images, nullity + rank = dimension', 'duration': 34.461, 'max_score': 33892.105, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI33892105.jpg'}, {'end': 33965.951, 'src': 'embed', 'start': 33940.97, 'weight': 2, 'content': [{'end': 33947.252, 'text': "And the answer is for every point in the range, an entire dimension's worth of points collapses to become that point.", 'start': 33940.97, 'duration': 6.282}, {'end': 33950.653, 'text': 'Rank plus nullity equals the dimension of the domain.', 'start': 33947.832, 'duration': 2.821}, {'end': 33954.902, 'text': 'So, of course, the check-in has some examples of that.', 'start': 33951.96, 'duration': 2.942}, {'end': 33956.003, 'text': 'And here we go.', 'start': 33955.263, 'duration': 0.74}, {'end': 33958.825, 'text': 'So two different homomorphisms.', 'start': 33956.123, 'duration': 2.702}, {'end': 33962.688, 'text': "And as usual, I'm not going to verify that the homomorphisms were kind of past that now.", 'start': 33958.925, 'duration': 3.763}, {'end': 33965.951, 'text': "And I'm looking to find the null space and the range space for each.", 'start': 33963.088, 'duration': 2.863}], 'summary': 'Homomorphisms collapse dimensions, rank+nullity=domain dimension.', 'duration': 24.981, 'max_score': 33940.97, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI33940970.jpg'}, {'end': 34173.277, 'src': 'embed', 'start': 34087.087, 'weight': 8, 'content': [{'end': 34096.194, 'text': "I'm not going to check linear independence or any of that stuff, but 0, 1, 0, and minus 2, 0, 1..", 'start': 34087.087, 'duration': 9.107}, {'end': 34096.714, 'text': "It's a basis.", 'start': 34096.194, 'duration': 0.52}, {'end': 34105.305, 'text': 'So because there are two things in the basis, the nullity is 2.', 'start': 34097.555, 'duration': 7.75}, {'end': 34107.646, 'text': 'Okay, so we have a good sense of what the null space is.', 'start': 34105.305, 'duration': 2.341}, {'end': 34108.626, 'text': "It's a plane.", 'start': 34107.986, 'duration': 0.64}, {'end': 34110.787, 'text': "Inside of R3, it's a plane.", 'start': 34108.986, 'duration': 1.801}, {'end': 34115.849, 'text': 'These two vectors whose bodies lie in the plane, not just the tip, the entire body lies in the plane.', 'start': 34111.367, 'duration': 4.482}, {'end': 34119.41, 'text': 'These two vectors form a basis for that plane.', 'start': 34116.169, 'duration': 3.241}, {'end': 34125.232, 'text': 'What about the range space? A range space of G.', 'start': 34120.371, 'duration': 4.861}, {'end': 34128.173, 'text': 'Sorry, I forgot to write the G.', 'start': 34125.232, 'duration': 2.941}, {'end': 34146.569, 'text': 'Now, because the nullity is 2, and the dimension of the domain is 3.', 'start': 34128.173, 'duration': 18.396}, {'end': 34158.055, 'text': "So we have that the rank of the map must be, you know, 2, 3, I mean it's a 1.", 'start': 34146.569, 'duration': 11.486}, {'end': 34165.459, 'text': 'So what that tells you is that the range space is just all multiples of x.', 'start': 34158.055, 'duration': 7.404}, {'end': 34173.277, 'text': "That's right, it's a one-dimensional space.", 'start': 34170.154, 'duration': 3.123}], 'summary': 'The nullity is 2, indicating a two-dimensional null space, and the range space is one-dimensional.', 'duration': 86.19, 'max_score': 34087.087, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI34087087.jpg'}, {'end': 34389.037, 'src': 'embed', 'start': 34359.244, 'weight': 3, 'content': [{'end': 34364.665, 'text': "it's clear that the range is a subspace, but the range has to be all of, or else it wouldn't have the right rank.", 'start': 34359.244, 'duration': 5.421}, {'end': 34366.526, 'text': 'Okay, very good.', 'start': 34365.906, 'duration': 0.62}, {'end': 34368.426, 'text': "Oops, put the cap on my pen so it doesn't run out.", 'start': 34366.546, 'duration': 1.88}, {'end': 34371.267, 'text': 'Okay, very, very good.', 'start': 34368.806, 'duration': 2.461}, {'end': 34378.369, 'text': 'Okay, so what I want to talk about today then is what I began discussing.', 'start': 34372.305, 'duration': 6.064}, {'end': 34381.892, 'text': 'We want to look at homomorphisms.', 'start': 34380.171, 'duration': 1.721}, {'end': 34389.037, 'text': "Homomorphisms are the point in this course where people start saying, I thought I was getting this and now I'm not so sure.", 'start': 34382.993, 'duration': 6.044}], 'summary': 'Discussion on homomorphisms and their significance in the course.', 'duration': 29.793, 'max_score': 34359.244, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI34359244.jpg'}, {'end': 34898.597, 'src': 'embed', 'start': 34864.447, 'weight': 4, 'content': [{'end': 34869.929, 'text': 'So it starts with red on the right-hand side and moving counterclockwise, moves through blue and violet.', 'start': 34864.447, 'duration': 5.482}, {'end': 34876.869, 'text': 'When you multiply by minus 1, now red is on the left-hand side, and it moves like so.', 'start': 34870.947, 'duration': 5.922}, {'end': 34898.597, 'text': 'So that is to say that the action of multiplying by minus x changes the orientation by which you traverse this circle from counterclockwise to clockwise.', 'start': 34880.19, 'duration': 18.407}], 'summary': 'Multiplying by -1 changes orientation from counterclockwise to clockwise.', 'duration': 34.15, 'max_score': 34864.447, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI34864447.jpg'}, {'end': 35157.7, 'src': 'embed', 'start': 35123.408, 'weight': 5, 'content': [{'end': 35128.63, 'text': "Here is a skew in the y direction, so I'm going to mix the x's and y's in this way.", 'start': 35123.408, 'duration': 5.222}, {'end': 35133.231, 'text': "Leave the x alone, mix the x's and y's in this way, and you see the same effect here.", 'start': 35129.29, 'duration': 3.941}, {'end': 35136.732, 'text': 'You see that something interesting is happening here.', 'start': 35133.511, 'duration': 3.221}, {'end': 35147.636, 'text': "What's happening here is that when we put x's down here, if x is not 0, and indeed x is not 0, then there's a pronounced effect.", 'start': 35138.593, 'duration': 9.043}, {'end': 35157.7, 'text': 'So if x is not zero, then you see that the point moves.', 'start': 35154.619, 'duration': 3.081}], 'summary': "Mixing x's and y's causes a pronounced effect when x is not zero.", 'duration': 34.292, 'max_score': 35123.408, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI35123408.jpg'}, {'end': 35320.348, 'src': 'embed', 'start': 35289.251, 'weight': 6, 'content': [{'end': 35290.572, 'text': 'Some points on the half circle are moving.', 'start': 35289.251, 'duration': 1.321}, {'end': 35301.334, 'text': "Another example of an action that we've seen for linear maps that we talked about, for example, when we did bases, is rotation.", 'start': 35295.529, 'duration': 5.805}, {'end': 35302.795, 'text': 'So here I did rotation.', 'start': 35301.674, 'duration': 1.121}, {'end': 35304.916, 'text': 'This is plain, simple rotation.', 'start': 35302.975, 'duration': 1.941}, {'end': 35306.337, 'text': 'Everybody rotates.', 'start': 35305.297, 'duration': 1.04}, {'end': 35316.785, 'text': 'You pick up the entire half circle and rotate everybody by whatever degrees, whatever angle theta is.', 'start': 35306.618, 'duration': 10.167}, {'end': 35320.348, 'text': 'No change of orientation here.', 'start': 35318.847, 'duration': 1.501}], 'summary': 'Points on half circle move with simple rotation, no change in orientation.', 'duration': 31.097, 'max_score': 35289.251, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI35289251.jpg'}, {'end': 35380.969, 'src': 'embed', 'start': 35345.612, 'weight': 7, 'content': [{'end': 35346.493, 'text': 'You can hardly see it.', 'start': 35345.612, 'duration': 0.881}, {'end': 35349.433, 'text': "It's on the x-axis, so it's basically gone.", 'start': 35346.573, 'duration': 2.86}, {'end': 35351.374, 'text': 'You can kind of see the colors a little bit.', 'start': 35349.954, 'duration': 1.42}, {'end': 35359.866, 'text': "Now, the map doesn't have to squeeze down to the x-axis.", 'start': 35355.805, 'duration': 4.061}, {'end': 35363.026, 'text': 'You can make it squeeze down to the line, for example, y equals 2x.', 'start': 35359.926, 'duration': 3.1}, {'end': 35364.867, 'text': 'So there we go.', 'start': 35364.206, 'duration': 0.661}, {'end': 35367.267, 'text': "I've projected to the line y equals 2x.", 'start': 35365.087, 'duration': 2.18}, {'end': 35373.008, 'text': 'So some points project up, some points project down, but they all project to the line y equals 2x.', 'start': 35368.307, 'duration': 4.701}, {'end': 35380.969, 'text': "And again, I'm not a very imaginative person.", 'start': 35378.929, 'duration': 2.04}], 'summary': 'Points projected to line y=2x on the x-axis.', 'duration': 35.357, 'max_score': 35345.612, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI35345612.jpg'}], 'start': 33774.734, 'title': 'Homomorphisms and linear maps', 'summary': 'Explores homomorphisms in real spaces, focusing on organizational structure, action on domains, and algebraic expressions, accompanied by examples, emphasizing the collapse of dimensions and the nullity plus rank equation. it also discusses linear maps in the plane, emphasizing their actions and transformations, with visualizations and examples depicting stretching, dilation, rotation, and orientation changes of shapes.', 'chapters': [{'end': 34046.423, 'start': 33774.734, 'title': 'Understanding homomorphisms in real spaces', 'summary': 'Explores the concept of homomorphisms, focusing on their organizational structure, action on domains, and algebraic expressions, with examples in real spaces, emphasizing the collapse of dimensions and the nullity plus rank equation.', 'duration': 271.689, 'highlights': ['The homomorphism organizes the domain into inverse images of elements from the range, with examples demonstrating how homomorphisms act in different cases.', 'The concept of homomorphism is illustrated using the example of the projection from R3 to R2, where the collapse of dimensions is explained in terms of rank plus nullity equaling the dimension of the domain.', 'The chapter delves into the algebraic expression of homomorphisms, showcasing their ability to divide the domain into inverse images of points in the range, emphasizing the nullity plus rank equation.', 'The process of finding the null space and range space for different homomorphisms is discussed, highlighting the practical application of understanding homomorphisms in various scenarios.']}, {'end': 34427.68, 'start': 34046.423, 'title': 'Homomorphisms and null space', 'summary': 'Discusses the null space and range space of linear transformations and defines the nullity and rank, with examples showcasing the dimensions of these spaces and their basis.', 'duration': 381.257, 'highlights': ['The nullity of the first map is 2, leading to a plane as the null space in R3, while the range space is a one-dimensional space consisting of all multiples of x.', 'The nullity of the second map is 0, resulting in a trivial subspace with the basis being the empty sequence, and the range space being the line y equals 3x in the xy plane.', 'The chapter emphasizes the study of homomorphisms with various examples and highlights the importance of using numbers to understand abstract concepts.']}, {'end': 34783.234, 'start': 34427.74, 'title': 'Understanding linear maps in the plane', 'summary': 'Discusses how linear maps act in the plane, particularly emphasizing that under a transformation from rn to rn, lines through the origin map to lines through the origin, and to understand the action of a map from the plane to the plane, it suffices to understand what it does to lines through the origin.', 'duration': 355.494, 'highlights': ['Under a transformation from Rn to Rn, lines through the origin map to lines through the origin.', 'To understand what a map from the plane to the plane does to plane elements, it suffices to understand what it does to lines to the origin.', 'To understand what t does on a line through the origin, fix the line y equals 2x.']}, {'end': 35442.02, 'start': 34785.273, 'title': 'Linear maps and transformations', 'summary': 'Introduces linear maps and transformations in the plane, illustrating how they can stretch, dilate, rotate, and change the orientation of shapes, such as semicircles and ellipses, through examples and visualizations.', 'duration': 656.747, 'highlights': ['The action of multiplying by a negative number changes the orientation by which the circle is traversed from counterclockwise to clockwise, as demonstrated by multiplying by -1.', 'Changing the y-coordinate by a factor of 3 results in a dilation of the shape by a factor of 3, effectively stretching it, as depicted when y goes to 3y.', "Mixing x's and y's in the linear transformation causes a shear effect, visually demonstrated by leaning on a deck of cards to create a skew, with the top moving more than the bottom.", 'Rotation as a linear transformation is illustrated by rotating the entire half circle uniformly by a specified angle, demonstrating a consistent and uniform change in orientation.', 'Projection from the plane to a line, such as y = 2x, showcases how points are projected to the specified line, effectively mapping the points onto the line in a consistent manner.']}], 'duration': 1667.286, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI33774734.jpg', 'highlights': ['The chapter delves into the algebraic expression of homomorphisms, showcasing their ability to divide the domain into inverse images of points in the range, emphasizing the nullity plus rank equation.', 'The homomorphism organizes the domain into inverse images of elements from the range, with examples demonstrating how homomorphisms act in different cases.', 'The concept of homomorphism is illustrated using the example of the projection from R3 to R2, where the collapse of dimensions is explained in terms of rank plus nullity equaling the dimension of the domain.', 'The process of finding the null space and range space for different homomorphisms is discussed, highlighting the practical application of understanding homomorphisms in various scenarios.', 'The action of multiplying by a negative number changes the orientation by which the circle is traversed from counterclockwise to clockwise, as demonstrated by multiplying by -1.', "Mixing x's and y's in the linear transformation causes a shear effect, visually demonstrated by leaning on a deck of cards to create a skew, with the top moving more than the bottom.", 'Rotation as a linear transformation is illustrated by rotating the entire half circle uniformly by a specified angle, demonstrating a consistent and uniform change in orientation.', 'Projection from the plane to a line, such as y = 2x, showcases how points are projected to the specified line, effectively mapping the points onto the line in a consistent manner.', 'The nullity of the second map is 0, resulting in a trivial subspace with the basis being the empty sequence, and the range space being the line y equals 3x in the xy plane.', 'The nullity of the first map is 2, leading to a plane as the null space in R3, while the range space is a one-dimensional space consisting of all multiples of x.']}, {'end': 37857.434, 'segs': [{'end': 35470.224, 'src': 'embed', 'start': 35442.421, 'weight': 0, 'content': [{'end': 35449.005, 'text': "So not only do you see dilation, that you see stretching, you see some kind of rotation, but you'll also see orientation reversal.", 'start': 35442.421, 'duration': 6.584}, {'end': 35451.846, 'text': 'So 1, 2, 3, 4 is actually a pretty good example.', 'start': 35449.045, 'duration': 2.801}, {'end': 35459.451, 'text': 'Okay, so again, what is happening here is that we are trying to find different ways to look at homomorphisms.', 'start': 35453.587, 'duration': 5.864}, {'end': 35464.536, 'text': 'what is it that homomorphisms can do and get some picture in our mind,', 'start': 35460.29, 'duration': 4.246}, {'end': 35470.224, 'text': 'some sort of framework or schema on which we can hang new homomorphism thoughts?', 'start': 35464.536, 'duration': 5.688}], 'summary': 'Homomorphisms exhibit dilation, stretching, rotation, and orientation reversal, providing a framework for new thoughts.', 'duration': 27.803, 'max_score': 35442.421, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI35442421.jpg'}, {'end': 35903.621, 'src': 'embed', 'start': 35874.69, 'weight': 1, 'content': [{'end': 35885.858, 'text': "So we've already seen the result that I hope I emphasized is key, that the action of a linear map is determined by its action on a basis.", 'start': 35874.69, 'duration': 11.168}, {'end': 35890.821, 'text': 'That if you know what h does to beta 1 through beta n, then you know what h does to any v.', 'start': 35886.178, 'duration': 4.643}, {'end': 35897.975, 'text': 'And we, of course, had a term for how do you get from the betas to V.', 'start': 35893.85, 'duration': 4.125}, {'end': 35903.621, 'text': 'We call that extending the action of the map linearly from the basis to the entire domain.', 'start': 35897.975, 'duration': 5.646}], 'summary': "Linear map's action is determined by its action on a basis.", 'duration': 28.931, 'max_score': 35874.69, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI35874690.jpg'}, {'end': 36482.974, 'src': 'embed', 'start': 36455.302, 'weight': 3, 'content': [{'end': 36458.083, 'text': 'What is the action of the map on the second basis element? There it is.', 'start': 36455.302, 'duration': 2.781}, {'end': 36460.764, 'text': 'So I took the first basis element and projected it.', 'start': 36458.463, 'duration': 2.301}, {'end': 36463.145, 'text': 'I took the second basis element and projected it.', 'start': 36461.125, 'duration': 2.02}, {'end': 36464.866, 'text': 'I followed this formula.', 'start': 36463.746, 'duration': 1.12}, {'end': 36471.949, 'text': 'Once you know what is the action of the map on the first basis element, represent it with respect to the second basis.', 'start': 36466.347, 'duration': 5.602}, {'end': 36474.79, 'text': 'Represent it with respect to D.', 'start': 36472.289, 'duration': 2.501}, {'end': 36475.691, 'text': 'And likewise down here.', 'start': 36474.79, 'duration': 0.901}, {'end': 36482.974, 'text': 'Once you know the action of the map on the second basis element, represent it with respect to the output basis.', 'start': 36476.031, 'duration': 6.943}], 'summary': 'Discussing the action of a map on basis elements and representing it with respect to different bases.', 'duration': 27.672, 'max_score': 36455.302, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI36455302.jpg'}, {'end': 36526.602, 'src': 'embed', 'start': 36499.874, 'weight': 2, 'content': [{'end': 36504.978, 'text': 'So you write them down in some organized fashion, and the kind of natural organized fashion is to write them next to each other.', 'start': 36499.874, 'duration': 5.104}, {'end': 36509.561, 'text': 'So we get four numbers there, and those four numbers somehow represent this map.', 'start': 36505.998, 'duration': 3.563}, {'end': 36517.518, 'text': 'What happens if you change the basis? So I took the same map, changed the basis.', 'start': 36513.516, 'duration': 4.002}, {'end': 36519.179, 'text': 'This time I changed it to an easier one.', 'start': 36517.698, 'duration': 1.481}, {'end': 36522.741, 'text': "I've been kind of talking up how a generic one makes a better first example.", 'start': 36519.219, 'duration': 3.522}, {'end': 36526.602, 'text': 'So my second example is to take an easy basis.', 'start': 36523.061, 'duration': 3.541}], 'summary': 'Exploring map representation using four numbers and changing the basis for an easier example.', 'duration': 26.728, 'max_score': 36499.874, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI36499874.jpg'}, {'end': 37482.337, 'src': 'embed', 'start': 37454.417, 'weight': 4, 'content': [{'end': 37458.579, 'text': 'You take the second row, combine it with the single column, you get the second entry.', 'start': 37454.417, 'duration': 4.162}, {'end': 37467.022, 'text': 'Finally, down to the mth row, combine it with the single column, and you get the mth entry.', 'start': 37458.619, 'duration': 8.403}, {'end': 37472.144, 'text': 'So this is the theorem that we just applied to do the check-in.', 'start': 37468.82, 'duration': 3.324}, {'end': 37474.427, 'text': 'So just skip past here.', 'start': 37472.965, 'duration': 1.462}, {'end': 37478.693, 'text': 'We defined the matrix vector product to be that operation.', 'start': 37475.609, 'duration': 3.084}, {'end': 37482.337, 'text': 'This is the natural way to combine matrices and vectors.', 'start': 37478.973, 'duration': 3.364}], 'summary': 'Matrix-vector product theorem applied for combining matrices and vectors.', 'duration': 27.92, 'max_score': 37454.417, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI37454417.jpg'}], 'start': 35442.421, 'title': 'Linear maps and matrix computation', 'summary': 'Explores homomorphisms, linear transformations, and computational schemes using matrices, discussing capabilities such as dilation, rotation, orientation reversal, and specific calculations including null space, nullity, rank. it also introduces the computation scheme for linear maps, representation with matrices, effect calculation, and combination of matrix and column vector representation. additionally, it discusses matrix representation with respect to basis, computation of matrix vector multiplication, and the process of representing a map with respect to different bases, emphasizing the importance of organized number combinations. moreover, it covers the computation of arbitrary maps using a computational scheme, application of the matrix-vector product theorem, and the concept of representing functions using matrices, highlighting the importance of matching matrix and vector sizes.', 'chapters': [{'end': 35852.08, 'start': 35442.421, 'title': 'Homomorphisms and linear transformations', 'summary': 'Explores the properties of homomorphisms and linear transformations, discussing their capabilities such as dilation, rotation, orientation reversal, and computational schemes using matrices, while also delving into specific calculations such as finding null space, nullity, and rank.', 'duration': 409.659, 'highlights': ['Homomorphisms capabilities include dilation, stretching, rotation, and orientation reversal.', 'Upcoming focus on computational schemes for representing homomorphisms using matrices.', 'Detailed explanation and calculation of null space, nullity, and rank for the derivative map from p3 to p2.']}, {'end': 36276.718, 'start': 35853.261, 'title': 'Linear maps computation', 'summary': 'Introduces the scheme for computing with linear maps, emphasizing the representation of linear maps with matrices, the calculation of the effect of the map on an arbitrary v from the domain, and the definition of the combination of the matrix and column vector representation.', 'duration': 423.457, 'highlights': ['The representation of linear maps with matrices is emphasized, showing that the action of a linear map is determined by its action on a basis.', 'The calculation of the effect of the map on an arbitrary V from the domain is demonstrated through the combination of matrix and column vector representation.', 'The definition of the combination of the matrix and column vector representation is explained as the result of how the elements combine, following the distributive law and factoring out.']}, {'end': 36913.25, 'start': 36276.718, 'title': 'Matrix representation and computation', 'summary': 'Discusses the representation of linear maps with respect to domain and co-domain basis, emphasizing the computation of matrix vector multiplication and how it enables the computation of the effect of a linear map. it also covers the definition of matrix vector multiplication and highlights the importance of organized number combination in defining linear maps.', 'duration': 636.532, 'highlights': ['The chapter discusses the representation of linear maps with respect to domain and co-domain basis, emphasizing the computation of matrix vector multiplication and how it enables the computation of the effect of a linear map.', 'It covers the definition of matrix vector multiplication, which involves combining the numbers in the matrix with the numbers for the representation of the vector to compute the effect of a linear map.', 'The importance of organized number combination in defining linear maps is highlighted, emphasizing that the definitions are not arbitrary and are based on observing how different parts combine to give the desired representation.']}, {'end': 37373.855, 'start': 36914.019, 'title': 'Matrix representation with respect to b and d', 'summary': 'Illustrates the process of representing a map with respect to two different bases, emphasizing the importance of not making the examples too easy or too hard, and demonstrates the computation of the representation of a map and a vector with respect to a given basis, ultimately showing the representation with respect to d of the result.', 'duration': 459.836, 'highlights': ['The process of representing a map with respect to two different bases is illustrated, emphasizing the importance of not making the examples too easy or too hard.', 'The computation of the representation of a map and a vector with respect to a given basis is demonstrated, showing the representation with respect to D of the result.']}, {'end': 37857.434, 'start': 37373.857, 'title': 'Matrix and vector computation', 'summary': 'Covers the computation of arbitrary maps using a computational scheme, emphasizing the application of the matrix-vector product theorem and the concept of representing functions using matrices. the representation of a map and vector can be combined using the dot product operation to obtain the output vector. the matrix-vector product is used to represent the action of a linear map, and the importance of matching the sizes of matrices and vectors is highlighted.', 'duration': 483.577, 'highlights': ['The representation of a map and vector can be combined using the dot product operation, as per the matrix-vector product theorem.', 'The matrix-vector product is used to represent the action of a linear map, enabling the combination of the representation of the map and the input vector to produce the output vector.', 'The importance of matching the sizes of matrices and vectors is emphasized, as the product of matrices and vectors is defined only when the sizes match.', 'An example of projection onto the x-axis is demonstrated, showcasing the representation of the input vector with respect to the domain basis and its combination to obtain the representation with respect to the co-domain basis.', 'The emphasis is on the functions as the central object of study, with matrices representing those functions.']}], 'duration': 2415.013, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI35442421.jpg', 'highlights': ['Homomorphisms capabilities include dilation, stretching, rotation, and orientation reversal.', 'Representation of linear maps with matrices is emphasized, showing the action of a linear map on a basis.', 'Importance of organized number combination in defining linear maps is highlighted.', 'Process of representing a map with respect to two different bases is illustrated.', 'Matrix-vector product theorem enables the combination of map and input vector to produce the output vector.']}, {'end': 39682.311, 'segs': [{'end': 37926.656, 'src': 'embed', 'start': 37902.842, 'weight': 0, 'content': [{'end': 37909.926, 'text': "So we're giving the function primary, trying to help a person build the right mental scaffolding in their mind.", 'start': 37902.842, 'duration': 7.084}, {'end': 37911.407, 'text': 'What comes first? What comes second?', 'start': 37909.986, 'duration': 1.421}, {'end': 37917.771, 'text': "And the point of view that we're presenting is that what comes first is functions of a certain kind linear functions, homomorphisms.", 'start': 37911.988, 'duration': 5.783}, {'end': 37926.656, 'text': 'And what comes second is that the matrices and column vectors represent those in the way that the receipts, the chits that I handed in,', 'start': 37918.391, 'duration': 8.265}], 'summary': 'Teaching linear functions as primary and matrices as secondary in mental scaffolding.', 'duration': 23.814, 'max_score': 37902.842, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI37902842.jpg'}, {'end': 38526.91, 'src': 'embed', 'start': 38502.287, 'weight': 1, 'content': [{'end': 38508.313, 'text': 'Rank The rank of a matrix is equal to the rank of any map it represents.', 'start': 38502.287, 'duration': 6.026}, {'end': 38512.436, 'text': "So you'll remember that the rank is the dimension of the range.", 'start': 38508.693, 'duration': 3.743}, {'end': 38515.039, 'text': 'The rank of a map is the dimension of the range.', 'start': 38512.537, 'duration': 2.502}, {'end': 38521.545, 'text': "So this says the rank of a matrix you do Gauss's method and how many rows non-zero rows remain.", 'start': 38515.399, 'duration': 6.146}, {'end': 38524.528, 'text': "that's equal to the rank of the map that it represents.", 'start': 38521.545, 'duration': 2.983}, {'end': 38526.91, 'text': "what's the dimension of the range?", 'start': 38524.528, 'duration': 2.382}], 'summary': 'Matrix rank equals map rank, both equal range dimension.', 'duration': 24.623, 'max_score': 38502.287, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI38502287.jpg'}, {'end': 38767.502, 'src': 'embed', 'start': 38740.937, 'weight': 2, 'content': [{'end': 38744.718, 'text': 'Remember, we did some reasonably complicated calculations in the first section of the third chapter.', 'start': 38740.937, 'duration': 3.781}, {'end': 38747.159, 'text': 'You want to know if your map is an isomorphism.', 'start': 38745.019, 'duration': 2.14}, {'end': 38750.8, 'text': 'Here is a strictly computational way requires no insight.', 'start': 38747.459, 'duration': 3.341}, {'end': 38752.561, 'text': "You don't have to do the hard part about onto's.", 'start': 38750.82, 'duration': 1.741}, {'end': 38758.108, 'text': 'You represent it with respect to some basis for the domain and codomain.', 'start': 38753.101, 'duration': 5.007}, {'end': 38759.49, 'text': "You do Gauss's method.", 'start': 38758.369, 'duration': 1.121}, {'end': 38767.502, 'text': "Do you have a square matrix that has no zero rows? Then it's an isomorph.", 'start': 38759.971, 'duration': 7.531}], 'summary': "Use gauss's method to check isomorphism of square matrix.", 'duration': 26.565, 'max_score': 38740.937, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI38740937.jpg'}, {'end': 39325.622, 'src': 'embed', 'start': 39301.29, 'weight': 3, 'content': [{'end': 39308.572, 'text': "Since the output space, the codomain space has dimension 2, because that's the number of rows in the starting matrix.", 'start': 39301.29, 'duration': 7.282}, {'end': 39313.933, 'text': 'since the codomain space has dimension 2, the rank of the map, it covers the entire codomain space.', 'start': 39308.572, 'duration': 5.361}, {'end': 39314.894, 'text': "That's what onto means.", 'start': 39313.973, 'duration': 0.921}, {'end': 39319.298, 'text': 'The rank of the map is 2.', 'start': 39315.154, 'duration': 4.144}, {'end': 39324.121, 'text': "For each a and b, there's more than one triple, x, y, z.", 'start': 39319.298, 'duration': 4.823}, {'end': 39325.622, 'text': 'So the map is not one-to-one.', 'start': 39324.121, 'duration': 1.501}], 'summary': 'Map has rank 2, not one-to-one, covers entire codomain space', 'duration': 24.332, 'max_score': 39301.29, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI39301290.jpg'}, {'end': 39428.919, 'src': 'embed', 'start': 39394.952, 'weight': 4, 'content': [{'end': 39399.095, 'text': 'If you take a and b and c to be 0, then you get that x equals 0, y equals 0.', 'start': 39394.952, 'duration': 4.143}, {'end': 39402.483, 'text': 'So the null space is trivial, so h is 1 to 1.', 'start': 39399.095, 'duration': 3.388}, {'end': 39410.75, 'text': 'The range, on the other hand, contains those output vectors whose representations satisfy that 0 equals a minus 2b plus c.', 'start': 39402.483, 'duration': 8.267}, {'end': 39416.074, 'text': 'So h is not onto only some of the codomain spaces.', 'start': 39410.75, 'duration': 5.324}, {'end': 39428.919, 'text': 'elements are in the range, namely those whose representations satisfy that, with respect to this basis, satisfy that a minus 2b plus c equals 0..', 'start': 39416.074, 'duration': 12.845}], 'summary': 'Null space is trivial, range is not onto, h is 1 to 1.', 'duration': 33.967, 'max_score': 39394.952, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI39394952.jpg'}], 'start': 37859.175, 'title': 'Linear algebra concepts', 'summary': 'Emphasizes the importance of understanding functions as the primary concept in linear algebra, discusses matrix representations of linear maps, representation of linear maps using matrices, and explains linear maps, non-singular matrices, and isomorphisms, with examples demonstrating computations and representations.', 'chapters': [{'end': 37917.771, 'start': 37859.175, 'title': 'Understanding functions and matrices', 'summary': 'Emphasizes the importance of understanding functions as the primary concept, rather than matrices, in linear algebra, highlighting the role of functions in computation and mental scaffolding.', 'duration': 58.596, 'highlights': ['The chapter stresses the importance of understanding functions as the primary concept in linear algebra, with matrices serving as representations of those functions.', 'It emphasizes the role of functions in computation, particularly in the context of matrix vector multiplication, as the real entities in the mathematical framework.', 'The emphasis is on helping individuals construct the correct mental framework, prioritizing functions over matrices, and comprehending the sequence of concepts in linear algebra.']}, {'end': 38324.327, 'start': 37918.391, 'title': 'Matrix representations and linear maps', 'summary': 'Discusses matrix representations of linear maps, illustrating how any matrix represents a linear map and demonstrating the concept through examples, including a verification that the map represented by a given matrix is linear.', 'duration': 405.936, 'highlights': ['Any matrix represents a linear map', 'Verification of linearity for a specific map represented by a given matrix', 'Explanation of matrix-vector multiplication', 'Incompatibility of matrix and vector dimensions']}, {'end': 38648.349, 'start': 38326.823, 'title': 'Understanding linear maps and matrix representations', 'summary': 'Discusses the representation of linear maps using matrices, emphasizing the dimensions, the zero map, and the rank of the matrix, and how they relate to the properties of the map, such as onto, one-to-one, and the dimension of the range.', 'duration': 321.526, 'highlights': ['The rank of a matrix is equal to the rank of any map it represents.', "The dimension of H's domain is equal to the number of columns in the matrix representing H, while the dimension of the codomain is equal to the number of rows.", 'A linear map represented by the matrix capital H is onto if, and only if, the rank of capital H equals the number of rows, and it is one-to-one if the rank equals the number of its columns.']}, {'end': 38972.156, 'start': 38648.349, 'title': 'Linear maps and isomorphisms', 'summary': "Explains the concept of linear maps, non-singular matrices, and isomorphisms, emphasizing the use of gauss's method to determine if a given map is an isomorphism, with examples demonstrating the computation of range space and null space for linear maps.", 'duration': 323.807, 'highlights': ["The chapter emphasizes the use of Gauss's method to determine if a given map is an isomorphism by checking for zero rows in the square matrix representation, providing a strictly computational approach that requires no insight.", "The concept of non-singular matrices is linked to the idea of non-singular linear maps, with a non-singular linear map being represented by a square matrix, and the connection to isomorphism is established, where a matrix represents isomorphism if and only if it's square and non-singular.", 'The computation of range space and null space for linear maps is demonstrated with examples, showcasing the determination of onto-ness and one-to-oneness for specific functions, with explanations of why certain functions are not onto or one-to-one.']}, {'end': 39682.311, 'start': 38972.316, 'title': 'Matrix representation and calculations', 'summary': 'Covers the calculation and representation of vectors and matrices, including determining range space, null space, and onto and one-to-one properties. it also emphasizes the use of bases and representations to understand underlying spaces and functions.', 'duration': 709.995, 'highlights': ['The rank of the map is 2, covering the entire codomain space, making it onto.', 'The null space is a subspace of the domain, and it has a basis consisting of a single vector, resulting in a nullity of 1.', "The range is of dimension 2 with a basis of 3 + x and -1 + x^2, indicating the range's underlying space characteristics.", 'The null space is trivial, indicating that the map is one-to-one.']}], 'duration': 1823.136, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI37859175.jpg', 'highlights': ['The chapter stresses the importance of understanding functions as the primary concept in linear algebra, with matrices serving as representations of those functions.', 'The rank of a matrix is equal to the rank of any map it represents.', "The chapter emphasizes the use of Gauss's method to determine if a given map is an isomorphism by checking for zero rows in the square matrix representation, providing a strictly computational approach that requires no insight.", 'The rank of the map is 2, covering the entire codomain space, making it onto.', 'The null space is trivial, indicating that the map is one-to-one.']}, {'end': 41984.112, 'segs': [{'end': 39841.698, 'src': 'embed', 'start': 39777.896, 'weight': 0, 'content': [{'end': 39785.139, 'text': 'AB representing, with respect to D, some W.', 'start': 39777.896, 'duration': 7.243}, {'end': 39798.105, 'text': 'There is an AB, excuse me, there is an XY representing, with respect to B, some V.', 'start': 39785.139, 'duration': 12.966}, {'end': 39799.766, 'text': 'In short, this map is onto.', 'start': 39798.105, 'duration': 1.661}, {'end': 39812.179, 'text': 'And the range space of f is all of the codomain.', 'start': 39806.135, 'duration': 6.044}, {'end': 39818.383, 'text': 'The range space of f is all of the codomain.', 'start': 39816.522, 'duration': 1.861}, {'end': 39834.974, 'text': "The same calculation shows if you take a and b to be 0, then you're determining what's in the null space.", 'start': 39819.644, 'duration': 15.33}, {'end': 39841.698, 'text': "There's a unique solution.", 'start': 39840.677, 'duration': 1.021}], 'summary': 'Map is onto, range space is full, unique solution in null space.', 'duration': 63.802, 'max_score': 39777.896, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI39777896.jpg'}, {'end': 39946.879, 'src': 'embed', 'start': 39895.425, 'weight': 1, 'content': [{'end': 39907.443, 'text': 'so the null space here of f is the trivial subspace of the domain.', 'start': 39895.425, 'duration': 12.018}, {'end': 39911.326, 'text': 'So this map is one-to-one.', 'start': 39909.505, 'duration': 1.821}, {'end': 39923.194, 'text': "And since it's also onto, so it is an isomorphism.", 'start': 39918.331, 'duration': 4.863}, {'end': 39931.232, 'text': 'The map.', 'start': 39930.572, 'duration': 0.66}, {'end': 39932.733, 'text': 'that is well.', 'start': 39931.232, 'duration': 1.501}, {'end': 39942.817, 'text': "I haven't written down the map, but the map that is represented by this matrix with respect to these bases is an isomorphism from R2 to R2,", 'start': 39932.733, 'duration': 10.084}, {'end': 39945.478, 'text': 'and so I almost snuck it out there early.', 'start': 39942.817, 'duration': 2.661}, {'end': 39946.879, 'text': 'that makes it an automorphism.', 'start': 39945.478, 'duration': 1.401}], 'summary': 'The given map is an isomorphism from r2 to r2, making it an automorphism.', 'duration': 51.454, 'max_score': 39895.425, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI39895425.jpg'}, {'end': 41313.66, 'src': 'embed', 'start': 41264.047, 'weight': 3, 'content': [{'end': 41267.29, 'text': "The function operation we're going to do now is the composition of two functions.", 'start': 41264.047, 'duration': 3.243}, {'end': 41272.294, 'text': "So if you have f as a linear function, g as a linear function and it's legal to compose them,", 'start': 41267.73, 'duration': 4.564}, {'end': 41277.758, 'text': 'the domains and codomains match up then how do you represent f composed with g??', 'start': 41272.294, 'duration': 5.464}, {'end': 41286.962, 'text': 'Okay, so, another function operation besides scalar multiplication and addition is composition.', 'start': 41282.278, 'duration': 4.684}, {'end': 41289.904, 'text': 'And the composition of linear maps is linear.', 'start': 41287.482, 'duration': 2.422}, {'end': 41293.587, 'text': "We actually have already seen that, but it's not very hard to show.", 'start': 41290.184, 'duration': 3.403}, {'end': 41299.252, 'text': "So I have G composed with H, and I've arranged so that the domains and codomains match up.", 'start': 41294.168, 'duration': 5.084}, {'end': 41304.035, 'text': 'Specifically, the outputs of H and the inputs of G are the same.', 'start': 41299.592, 'duration': 4.443}, {'end': 41307.899, 'text': 'So G circle H.', 'start': 41306.818, 'duration': 1.081}, {'end': 41313.66, 'text': "on a combination is the combination of the G circle H's.", 'start': 41309.216, 'duration': 4.444}], 'summary': 'Linear function composition is discussed, showing g composed with h with matching domains and codomains.', 'duration': 49.613, 'max_score': 41264.047, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI41264047.jpg'}, {'end': 41965.678, 'src': 'embed', 'start': 41934.437, 'weight': 5, 'content': [{'end': 41935.558, 'text': 'Oh, I lost a minus sign.', 'start': 41934.437, 'duration': 1.121}, {'end': 41936.218, 'text': 'I got to go back.', 'start': 41935.598, 'duration': 0.62}, {'end': 41940.101, 'text': "But no, then, and this is not defined, but don't lose track.", 'start': 41936.478, 'duration': 3.623}, {'end': 41942.102, 'text': "You know, don't lose the forest for the trees.", 'start': 41940.141, 'duration': 1.961}, {'end': 41947.846, 'text': 'The composition of linear maps is represented by the matrix product of the representatives.', 'start': 41942.482, 'duration': 5.364}, {'end': 41951.969, 'text': 'That is to say, matrix multiplication is a somewhat peculiar definition.', 'start': 41948.146, 'duration': 3.823}, {'end': 41956.412, 'text': 'You say, so why are you doing this crazy way? And the answer is because it represents.', 'start': 41951.989, 'duration': 4.423}, {'end': 41958.793, 'text': 'The composition of the functions.', 'start': 41957.392, 'duration': 1.401}, {'end': 41965.678, 'text': 'So if you have some conception of what the functions are, this function rotates and then this function makes 3 bigger.', 'start': 41959.254, 'duration': 6.424}], 'summary': 'Matrix multiplication represents composition of linear maps.', 'duration': 31.241, 'max_score': 41934.437, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI41934437.jpg'}], 'start': 39684.443, 'title': 'Matrix operations and representations', 'summary': 'Covers topics such as isomorphism of r2, matrix operations and representations, matrix mapping and representations, function composition in linear algebra, and matrix composition and multiplication. it emphasizes fundamental theorems, matrix operations, and their reflection of underlying properties of maps.', 'chapters': [{'end': 39946.879, 'start': 39684.443, 'title': 'Isomorphism of r2', 'summary': 'Explains that the given map is onto and the range space of f is all of the codomain, making it an isomorphism from r2 to r2, with the null space of f being the trivial subspace of the domain.', 'duration': 262.436, 'highlights': ['The map is onto, and the range space of f is all of the codomain, making it an isomorphism from R2 to R2.', 'The null space of f is the trivial subspace of the domain, indicating that the map is one-to-one.', 'Given an AB from the codomain representing a vector from the codomain, there are no restrictions, and every AB is associated with some XY.', "If you take a and b to be 0, then you're determining what's in the null space, and there's a unique solution.", 'The map represented by this matrix with respect to these bases is an isomorphism from R2 to R2, making it an automorphism.']}, {'end': 40244.53, 'start': 39948.299, 'title': 'Matrix operations and representations', 'summary': 'Discusses the representations of maps using matrices, and the operations of matrix addition and scalar multiplication, emphasizing that these operations reflect the underlying properties of the maps and are derived from them.', 'duration': 296.231, 'highlights': ['The chapter emphasizes the use of matrices and representations as a way to compute vector spaces and maps and compute the relationships between vector spaces expressed by the maps.', 'It explains the operations of matrix addition and scalar multiplication, highlighting that these operations reflect the properties of the underlying maps and are derived from them.', 'The chapter discusses the computation of scalar multiplication and addition of matrices as reflecting the map facts, and that the definitions of these operations are derived from the properties of maps.', 'It emphasizes that the operations of matrix addition and scalar multiplication are straightforward and easy, derived from the properties of the underlying maps, and serve to represent function operations.']}, {'end': 40690.77, 'start': 40244.831, 'title': 'Matrix mapping and representations', 'summary': 'Discusses matrix mapping and representations, including the effect of scalar multiplication and addition on matrices, with examples using numbers like 6 and k, and emphasizes the fundamental theorem that the map r times h is represented by r times the matrix h and the map h plus g is represented by the matrix a plus g.', 'duration': 445.939, 'highlights': ['The matrix representing 6H multiplies all its components of the outcome by 6, resulting in a representation of 12v1 and 6v2, with the matrix having 12 and 6 for the outcome of 12v1 and 6v2.', 'The representation of the map F plus G is obtained by adding the representations of F and G together component-wise, following the same bases, and the scalar multiple of a matrix is the result of the entry-by-entry scalar multiplication, and the sum of two same-size matrices is their entry-by-entry sum.', 'The fundamental theorem states that the map R times H is represented by R times the matrix H and the map H plus G is represented by the matrix A plus G, and this theorem is true to the defined times and plus operations on matrices.']}, {'end': 41243.384, 'start': 40691.994, 'title': 'Matrix operations: addition and multiplication', 'summary': 'Introduces the operations of matrix scalar multiplication and matrix addition, emphasizing their reflection of the underlying math, leading up to the concept of matrix multiplication and its reflective nature. it also covers the representation of matrices with respect to different bases and their actions on input elements.', 'duration': 551.39, 'highlights': ['The chapter introduces the operations of matrix scalar multiplication and matrix addition, emphasizing their reflection of the underlying math.', 'The concept of matrix multiplication and its reflective nature is discussed, emphasizing its role in making sense of the underlying maps.', 'The representation of matrices with respect to different bases and their actions on input elements is demonstrated using examples.']}, {'end': 41445.024, 'start': 41243.664, 'title': 'Function composition in linear algebra', 'summary': 'Discusses the function operation of composition in linear algebra, specifically focusing on the composition of linear functions, showcasing their linearity, and using matrices to represent the composition, emphasizing the straightforwardness and practical approach of the concept.', 'duration': 201.36, 'highlights': ['The composition of linear maps is linear.', 'Utilizing matrices to represent the composition of linear functions.', 'Emphasizing the straightforwardness and practical approach of the concept.']}, {'end': 41984.112, 'start': 41447.826, 'title': 'Matrix composition and multiplication', 'summary': 'Explains the process of matrix composition and multiplication, detailing the computation involved and the significance of matrix multiplication in representing the composition of linear maps.', 'duration': 536.286, 'highlights': ['Matrix multiplication represents the composition of linear maps', 'Explanation of matrix multiplication computation', 'Significance of square matrices in matrix multiplication']}], 'duration': 2299.669, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/JnTa9XtvmfI/pics/JnTa9XtvmfI39684443.jpg', 'highlights': ['The map is onto, and the range space of f is all of the codomain, making it an isomorphism from R2 to R2.', 'The null space of f is the trivial subspace of the domain, indicating that the map is one-to-one.', 'The map represented by this matrix with respect to these bases is an isomorphism from R2 to R2, making it an automorphism.', 'The composition of linear maps is linear.', 'Utilizing matrices to represent the composition of linear functions.', 'Matrix multiplication represents the composition of linear maps']}], 'highlights': ['The importance of conditions 1 and 6, which ensure closure under vector addition and scalar multiplication, as they are the crux of the matter (relevance: 5)', "The chapter emphasizes the concept of subspaces within vector spaces, explaining that a subspace is a subset that's itself a vector space under the inherited operation, and provides examples of subspaces, including the collection of 2x2 matrices with specific limitations.", 'The chapter discusses representing polynomials with three tall vectors, facilitating easier understanding and manipulation of the data.', 'The chapter emphasizes the importance of having only trivial solutions to determine linear independence.', "The row rank and column rank of a matrix are equal, indicating the same number of linearly independent rows and columns, with the practical example illustrating the calculation of column rank through Gauss's method resulting in 3 linearly independent rows and hence a column rank of 3."]}