title

Vectors | Chapter 1, Essence of linear algebra

description

Beginning the linear algebra series with the basics.
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Correction: 6:52, the screen should show [x1, y1] + [x2, y2] = [x1+x2, y1+y2]
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{'title': 'Vectors | Chapter 1, Essence of linear algebra', 'heatmap': [{'end': 421.34, 'start': 405.726, 'weight': 1}], 'summary': "Chapter 1 of 'essence of linear algebra' introduces vectors from three perspectives - physics, computer science, and mathematics. it defines vectors as arrows with length and direction, ordered lists of numbers, and a generalization of both views. the chapter focuses on vector addition, multiplication by numbers, and the relationship between geometric and numerical representations.", 'chapters': [{'end': 575.997, 'segs': [{'end': 70.557, 'src': 'embed', 'start': 30.777, 'weight': 0, 'content': [{'end': 34.36, 'text': 'The physics student perspective is that vectors are arrows pointing in space.', 'start': 30.777, 'duration': 3.583}, {'end': 38.823, 'text': "What defines a given vector is its length and the direction it's pointing.", 'start': 34.9, 'duration': 3.923}, {'end': 43.166, 'text': "But as long as those two facts are the same, you can move it all around and it's still the same vector.", 'start': 39.183, 'duration': 3.983}, {'end': 50.011, 'text': 'Vectors that live in the flat plane are two-dimensional, and those sitting in broader space that you and I live in are three-dimensional.', 'start': 43.927, 'duration': 6.084}, {'end': 55.579, 'text': 'The computer science perspective is that vectors are ordered lists of numbers.', 'start': 51.775, 'duration': 3.804}, {'end': 62.748, 'text': "For example, let's say you were doing some analytics about house prices and the only features you cared about were square footage and price.", 'start': 56.24, 'duration': 6.508}, {'end': 68.635, 'text': 'You might model each house with a pair of numbers, the first indicating square footage and the second indicating price.', 'start': 63.248, 'duration': 5.387}, {'end': 70.557, 'text': 'Notice the order matters here.', 'start': 69.335, 'duration': 1.222}], 'summary': 'Vectors can be viewed as arrows in space with specific length and direction, or as ordered lists of numbers in computer science.', 'duration': 39.78, 'max_score': 30.777, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/fNk_zzaMoSs/pics/fNk_zzaMoSs30777.jpg'}, {'end': 160.55, 'src': 'embed', 'start': 124.71, 'weight': 4, 'content': [{'end': 129.774, 'text': "Given the geometric focus that I'm shooting for here, whenever I introduce a new topic involving vectors,", 'start': 124.71, 'duration': 5.064}, {'end': 137.541, 'text': 'I want you to first think about an arrow and specifically think about that arrow inside a coordinate system like the xy-plane,', 'start': 129.774, 'duration': 7.767}, {'end': 138.862, 'text': 'with its tail sitting at the origin.', 'start': 137.541, 'duration': 1.321}, {'end': 144.873, 'text': 'This is a little bit different from the physics student perspective, where vectors can freely sit anywhere they want in space.', 'start': 139.687, 'duration': 5.186}, {'end': 150.319, 'text': "In linear algebra, it's almost always the case that your vector will be rooted at the origin.", 'start': 145.413, 'duration': 4.906}, {'end': 158.348, 'text': "Then, once you understand a new concept in the context of arrows in space, we'll translate it over to the list of numbers, point of view,", 'start': 151.08, 'duration': 7.268}, {'end': 160.55, 'text': 'which we can do by considering the coordinates of the vector.', 'start': 158.348, 'duration': 2.202}], 'summary': 'In linear algebra, vectors are rooted at the origin, then translated to a list of numbers.', 'duration': 35.84, 'max_score': 124.71, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/fNk_zzaMoSs/pics/fNk_zzaMoSs124710.jpg'}, {'end': 429.625, 'src': 'heatmap', 'start': 405.726, 'weight': 1, 'content': [{'end': 412.169, 'text': 'In general, vector addition in this list of numbers conception looks like matching up their terms and adding each one together.', 'start': 405.726, 'duration': 6.443}, {'end': 418.319, 'text': 'The other fundamental vector operation is multiplication by a number.', 'start': 414.997, 'duration': 3.322}, {'end': 421.34, 'text': 'Now this is best understood just by looking at a few examples.', 'start': 418.799, 'duration': 2.541}, {'end': 429.625, 'text': "If you take the number two and multiply it by a given vector, it means you stretch out that vector so that it's two times as long as when you started.", 'start': 421.981, 'duration': 7.644}], 'summary': 'Vector addition involves matching up terms and adding them. multiplying a vector by 2 stretches it to twice its length.', 'duration': 23.899, 'max_score': 405.726, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/fNk_zzaMoSs/pics/fNk_zzaMoSs405726.jpg'}, {'end': 509.066, 'src': 'embed', 'start': 482.914, 'weight': 3, 'content': [{'end': 488.318, 'text': 'multiplying a given vector by a scalar means multiplying each one of those components by that scalar.', 'start': 482.914, 'duration': 5.404}, {'end': 499.256, 'text': "You'll see in the following videos what I mean when I say that linear algebra topics tend to revolve around these two fundamental operations vector addition and scalar multiplication.", 'start': 490.506, 'duration': 8.75}, {'end': 507.445, 'text': "And I'll talk more in the last video about how and why the mathematician thinks only about these operations, independent and abstracted away from,", 'start': 499.976, 'duration': 7.469}, {'end': 509.066, 'text': 'however you choose to represent vectors.', 'start': 507.445, 'duration': 1.621}], 'summary': 'Linear algebra revolves around vector addition and scalar multiplication.', 'duration': 26.152, 'max_score': 482.914, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/fNk_zzaMoSs/pics/fNk_zzaMoSs482914.jpg'}], 'start': 11.313, 'title': 'Understanding vectors in linear algebra', 'summary': "Introduces the three perspectives on vectors - physics, computer science, and mathematician's perspective - defining vectors as arrows with length and direction, ordered lists of numbers, and a generalization of both views. it also focuses on vector addition, multiplication by numbers, and the relationship between geometric and numerical representations.", 'chapters': [{'end': 89.79, 'start': 11.313, 'title': 'Understanding vectors in linear algebra', 'summary': "Introduces the three perspectives on vectors - physics student, computer science student, and mathematician's perspective - defining vectors as arrows with length and direction, ordered lists of numbers, and a generalization of both views.", 'duration': 78.477, 'highlights': ["The mathematician's perspective seeks to generalize both the physics student and computer science student perspectives on vectors, offering a broader understanding.", 'The computer science perspective defines vectors as ordered lists of numbers, exemplifying the use of vectors in analytics, such as modeling houses based on square footage and price.', 'The physics student perspective views vectors as arrows in space, defined by length and direction, with two-dimensional vectors in a flat plane and three-dimensional vectors in broader space.']}, {'end': 575.997, 'start': 89.79, 'title': 'Vectors in linear algebra', 'summary': 'Introduces the concept of vectors in linear algebra, focusing on vector addition, multiplication by numbers, and the relationship between geometric and numerical representations.', 'duration': 486.207, 'highlights': ['The concept of vectors involves sensible addition and multiplication operations, which play an important role throughout linear algebra, with the representation of vectors rooted at the origin being a pivotal concept.', 'Vectors are initially viewed as arrows in a coordinate system, with their representation then translated to lists of numbers, and in three dimensions, each vector is associated with an ordered triplet of numbers.', 'Vector addition involves moving the tail of the second vector to the tip of the first, while scalar multiplication stretches, squishes, or reverses the direction of a vector, and these fundamental operations are central in linear algebra.', 'The ability to translate between the geometric and numerical representations of vectors is emphasized, with linear algebra providing a powerful tool for visualizing data and manipulating space using numbers.', 'The chapter provides a foundational understanding of vectors and hints at upcoming concepts such as span, bases, and linear dependence in the next video.']}], 'duration': 564.684, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/fNk_zzaMoSs/pics/fNk_zzaMoSs11313.jpg', 'highlights': ["The mathematician's perspective generalizes the physics and computer science student perspectives on vectors.", 'The computer science perspective defines vectors as ordered lists of numbers, used in analytics.', 'The physics student perspective views vectors as arrows in space, defined by length and direction.', 'Vector addition and scalar multiplication are fundamental operations central in linear algebra.', 'Translating between geometric and numerical representations of vectors is emphasized.', 'Vectors are initially viewed as arrows in a coordinate system, then translated to lists of numbers.']}], 'highlights': ['Vector addition and scalar multiplication are fundamental operations central in linear algebra.', "The mathematician's perspective generalizes the physics and computer science student perspectives on vectors.", 'The computer science perspective defines vectors as ordered lists of numbers, used in analytics.', 'The physics student perspective views vectors as arrows in space, defined by length and direction.', 'Translating between geometric and numerical representations of vectors is emphasized.', 'Vectors are initially viewed as arrows in a coordinate system, then translated to lists of numbers.']}