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Abstract vector spaces | Chapter 16, Essence of linear algebra

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This is really the reason linear algebra is so powerful.
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{'title': 'Abstract vector spaces | Chapter 16, Essence of linear algebra', 'heatmap': [{'end': 117.513, 'start': 84.534, 'weight': 0.735}, {'end': 362.462, 'start': 350.661, 'weight': 0.722}, {'end': 564.286, 'start': 547.222, 'weight': 1}, {'end': 588.301, 'start': 572.028, 'weight': 0.706}, {'end': 667.665, 'start': 623.08, 'weight': 0.965}], 'summary': 'Explores the spatial essence of vectors in linear algebra, demonstrates how functions can be viewed as vectors, and covers function spaces, polynomials, basis functions, and their relation to derivatives as an infinite matrix, providing a deeper insight into the foundational nature of linear algebra.', 'chapters': [{'end': 122.455, 'segs': [{'end': 55.446, 'src': 'embed', 'start': 17.202, 'weight': 1, 'content': [{'end': 21.966, 'text': "I'd like to revisit a deceptively simple question that I asked in the very first video of this series.", 'start': 17.202, 'duration': 4.764}, {'end': 23.607, 'text': 'What are vectors??', 'start': 22.626, 'duration': 0.981}, {'end': 26.73, 'text': 'Is a two-dimensional vector, for example,', 'start': 24.548, 'duration': 2.182}, {'end': 37.679, 'text': 'fundamentally an arrow on a flat plane that we can describe with coordinates for convenience? Or is it fundamentally that pair of real numbers which is just nicely visualized as an arrow on a flat plane??', 'start': 26.73, 'duration': 10.949}, {'end': 41.281, 'text': 'Or are both of these just manifestations of something deeper??', 'start': 38.599, 'duration': 2.682}, {'end': 48.522, 'text': 'On the one hand, defining vectors as primarily being a list of numbers feels clear cut and unambiguous.', 'start': 42.338, 'duration': 6.184}, {'end': 55.446, 'text': 'It makes things like four dimensional vectors or 100 dimensional vectors sound like real concrete ideas that you can actually work with.', 'start': 49.042, 'duration': 6.404}], 'summary': 'Revisiting the concept of vectors and their fundamental nature, dimensions, and visual representation.', 'duration': 38.244, 'max_score': 17.202, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TgKwz5Ikpc8/pics/TgKwz5Ikpc817202.jpg'}, {'end': 122.455, 'src': 'heatmap', 'start': 78.408, 'weight': 0, 'content': [{'end': 83.613, 'text': 'And the coordinates are actually somewhat arbitrary, depending on what you happen to choose as your basis vectors.', 'start': 78.408, 'duration': 5.205}, {'end': 90.597, 'text': 'Core topics in linear algebra like determinants and eigenvectors seem indifferent to your choice of coordinate systems.', 'start': 84.534, 'duration': 6.063}, {'end': 99.36, 'text': 'The determinant tells you how much a transformation scales areas, and eigenvectors are the ones that stay on their own span during a transformation.', 'start': 91.357, 'duration': 8.003}, {'end': 108.204, 'text': 'But both of these properties are inherently spatial and you can freely change your coordinate system without changing the underlying values of either one.', 'start': 100.061, 'duration': 8.143}, {'end': 117.513, 'text': 'But if vectors are not fundamentally lists of real numbers and if their underlying essence is something more spatial,', 'start': 111.089, 'duration': 6.424}, {'end': 122.455, 'text': 'that just begs the question of what mathematicians mean when they use a word like space or spatial.', 'start': 117.513, 'duration': 4.942}], 'summary': 'In linear algebra, determinants and eigenvectors are not affected by the choice of coordinate system.', 'duration': 44.047, 'max_score': 78.408, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TgKwz5Ikpc8/pics/TgKwz5Ikpc878408.jpg'}], 'start': 17.202, 'title': 'Understanding vectors in linear algebra', 'summary': 'Revisits the fundamental nature of vectors in linear algebra, exploring their spatial essence, and challenging the conventional understanding, providing a deeper insight into their nature.', 'chapters': [{'end': 122.455, 'start': 17.202, 'title': 'Understanding vectors in linear algebra', 'summary': 'Revisits the fundamental nature of vectors in linear algebra, exploring whether they are primarily a list of numbers or something deeper, and delves into the spatial essence of vectors, challenging the conventional understanding of vectors as merely coordinates.', 'duration': 105.253, 'highlights': ['Vectors can be perceived as either a list of numbers or a deeper spatial concept, challenging the conventional understanding of vectors as coordinates.', 'Core topics like determinants and eigenvectors in linear algebra are indifferent to the choice of coordinate systems, showcasing the spatial essence of vectors.', 'The chapter questions the fundamental nature of vectors and their underlying spatial essence, challenging the conventional perception of vectors as lists of real numbers.']}], 'duration': 105.253, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TgKwz5Ikpc8/pics/TgKwz5Ikpc817202.jpg', 'highlights': ['Core topics like determinants and eigenvectors in linear algebra are indifferent to the choice of coordinate systems, showcasing the spatial essence of vectors.', 'Vectors can be perceived as either a list of numbers or a deeper spatial concept, challenging the conventional understanding of vectors as coordinates.', 'The chapter questions the fundamental nature of vectors and their underlying spatial essence, challenging the conventional perception of vectors as lists of real numbers.']}, {'end': 416.048, 'segs': [{'end': 164.386, 'src': 'embed', 'start': 123.416, 'weight': 0, 'content': [{'end': 124.837, 'text': 'To build up to where this is going.', 'start': 123.416, 'duration': 1.421}, {'end': 130.24, 'text': "I'd actually like to spend the bulk of this video talking about something which is neither an arrow nor a list of numbers,", 'start': 124.837, 'duration': 5.403}, {'end': 133.181, 'text': 'but also has vector-ish qualities functions.', 'start': 130.24, 'duration': 2.941}, {'end': 137.744, 'text': "You see, there's a sense in which functions are actually just another type of vector.", 'start': 134.082, 'duration': 3.662}, {'end': 142.346, 'text': 'In the same way that you can add two vectors together.', 'start': 139.764, 'duration': 2.582}, {'end': 148.472, 'text': "there's also a sensible notion for adding two functions, f and g, to get a new function f plus g.", 'start': 142.346, 'duration': 6.126}, {'end': 153.096, 'text': "It's one of those things where you kind of already know what it's gonna be, but actually phrasing it is a mouthful.", 'start': 148.472, 'duration': 4.624}, {'end': 157.14, 'text': 'The output of this new function at any given input.', 'start': 153.957, 'duration': 3.183}, {'end': 164.386, 'text': 'like negative four is the sum of the outputs of f and g when you evaluate them each at that same input, negative four', 'start': 157.14, 'duration': 7.246}], 'summary': 'Functions can be viewed as vectors and added together, similar to vector addition.', 'duration': 40.97, 'max_score': 123.416, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TgKwz5Ikpc8/pics/TgKwz5Ikpc8123416.jpg'}, {'end': 376.753, 'src': 'heatmap', 'start': 350.661, 'weight': 0.722, 'content': [{'end': 355.944, 'text': 'is that a linear transformation is completely described by where it takes the basis vectors.', 'start': 350.661, 'duration': 5.283}, {'end': 362.462, 'text': 'Since any vector can be expressed by scaling and adding the basis vectors in some way,', 'start': 357.819, 'duration': 4.643}, {'end': 370.268, 'text': 'finding the transformed version of a vector comes down to scaling and adding the transformed versions of the basis vectors in that same way.', 'start': 362.462, 'duration': 7.806}, {'end': 376.753, 'text': "As you'll see in just a moment, this is as true for functions as it is for arrows.", 'start': 372.55, 'duration': 4.203}], 'summary': 'Linear transformation is completely described by basis vectors, applicable to functions as well.', 'duration': 26.092, 'max_score': 350.661, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TgKwz5Ikpc8/pics/TgKwz5Ikpc8350661.jpg'}, {'end': 416.048, 'src': 'embed', 'start': 362.462, 'weight': 2, 'content': [{'end': 370.268, 'text': 'finding the transformed version of a vector comes down to scaling and adding the transformed versions of the basis vectors in that same way.', 'start': 362.462, 'duration': 7.806}, {'end': 376.753, 'text': "As you'll see in just a moment, this is as true for functions as it is for arrows.", 'start': 372.55, 'duration': 4.203}, {'end': 384.758, 'text': 'For example, calculus students are always using the fact that the derivative is additive and has the scaling property,', 'start': 378.414, 'duration': 6.344}, {'end': 386.68, 'text': "even if they haven't heard it phrased that way.", 'start': 384.758, 'duration': 1.922}, {'end': 396.556, 'text': "If you add two functions, then take the derivative, it's the same as first taking the derivative of each one separately, then adding the result.", 'start': 388.211, 'duration': 8.345}, {'end': 406.822, 'text': "Similarly, if you scale a function, then take the derivative, it's the same as first taking the derivative, then scaling the result.", 'start': 400.198, 'duration': 6.624}, {'end': 416.048, 'text': "To really drill in the parallel, let's see what it might look like to describe the derivative with a matrix.", 'start': 410.705, 'duration': 5.343}], 'summary': 'Transformed vectors and functions follow scaling and additive properties, illustrated through derivatives in calculus.', 'duration': 53.586, 'max_score': 362.462, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TgKwz5Ikpc8/pics/TgKwz5Ikpc8362462.jpg'}], 'start': 123.416, 'title': 'Functions and linear transformations', 'summary': 'Explains how functions can be viewed as vectors, demonstrating addition and scaling akin to vectors, and explores linear transformations for functions with a focus on additive and scaling properties, and their application to derivatives and matrix vector multiplication.', 'chapters': [{'end': 214.231, 'start': 123.416, 'title': 'Functions as vectors', 'summary': 'Explains how functions can be viewed as vectors, demonstrating the addition and scaling of functions akin to vectors, with a comparison of their coordinate-wise operations and infinite coordinates.', 'duration': 90.815, 'highlights': ['Functions can be viewed as vectors, with the ability to add two functions together and scale a function by a real number, analogous to vector operations, demonstrating a sense of infinitely many coordinates.', 'The addition of two functions f and g results in a new function f plus g, where the output at any given input x is the sum of the values f of x plus g of x.', 'Similarly to adding vectors coordinate by coordinate, the value of the sum function at any given input x is the sum of the values f of x plus g of x.']}, {'end': 416.048, 'start': 214.231, 'title': 'Linear transformations for functions', 'summary': 'Explores the concept of linear transformations for functions, specifically focusing on the additive and scaling properties, and how they apply to derivatives in calculus and matrix vector multiplication.', 'duration': 201.817, 'highlights': ['The concept of linear transformations for functions is introduced, emphasizing the additive and scaling properties, which are essential for preserving vector addition and scalar multiplication.', 'The application of these properties to derivatives in calculus is highlighted, demonstrating that the derivative of a sum of functions equals the sum of their derivatives, and the derivative of a scaled function equals the scaled derivative.', 'The importance of basis vectors in completely describing linear transformations, and how it applies to both arrows and functions, is explained, laying the foundation for understanding matrix vector multiplication.']}], 'duration': 292.632, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TgKwz5Ikpc8/pics/TgKwz5Ikpc8123416.jpg', 'highlights': ['Functions can be viewed as vectors, with the ability to add and scale, analogous to vector operations.', 'The addition of two functions results in a new function f plus g, with output as the sum of values.', 'Linear transformations for functions emphasize additive and scaling properties, essential for preserving vector operations.', 'The application of properties to derivatives shows that the derivative of a sum equals the sum of derivatives.', 'Basis vectors are important in completely describing linear transformations, applicable to both arrows and functions.']}, {'end': 986.051, 'segs': [{'end': 512.967, 'src': 'embed', 'start': 482.189, 'weight': 0, 'content': [{'end': 487.495, 'text': 'Since our polynomials can have arbitrarily large degree, this set of basis functions is infinite.', 'start': 482.189, 'duration': 5.306}, {'end': 488.977, 'text': "But that's okay.", 'start': 488.216, 'duration': 0.761}, {'end': 494.123, 'text': "It just means that when we treat our polynomials as vectors, they're gonna have infinitely many coordinates.", 'start': 489.357, 'duration': 4.766}, {'end': 505.424, 'text': 'A polynomial like x squared plus three x plus five, for example, would be described with the coordinates five, three, one, then infinitely many zeros.', 'start': 495.58, 'duration': 9.844}, {'end': 512.967, 'text': "You'd read this as saying that it's five times the first basis function, plus three times that second basis function,", 'start': 506.104, 'duration': 6.863}], 'summary': 'Polynomials as vectors have infinitely many coordinates in the set of basis functions.', 'duration': 30.778, 'max_score': 482.189, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TgKwz5Ikpc8/pics/TgKwz5Ikpc8482189.jpg'}, {'end': 571.288, 'src': 'heatmap', 'start': 533.374, 'weight': 2, 'content': [{'end': 538.58, 'text': 'In general, since every individual polynomial has only finitely many terms,', 'start': 533.374, 'duration': 5.206}, {'end': 542.784, 'text': 'its coordinates will be some finite string of numbers with an infinite tail of zeros.', 'start': 538.58, 'duration': 4.204}, {'end': 555.284, 'text': "In this coordinate system, the derivative is described with an infinite matrix that's mostly full of zeros, but which has the positive integers,", 'start': 547.222, 'duration': 8.062}, {'end': 557.645, 'text': 'counting down on this offset diagonal.', 'start': 555.284, 'duration': 2.361}, {'end': 564.286, 'text': "I'll talk about how you could find this matrix in just a moment, but the best way to get a feel for it is to just watch it in action.", 'start': 558.385, 'duration': 5.901}, {'end': 571.288, 'text': 'Take the coordinates representing the polynomial x cubed plus five x squared plus four x plus five.', 'start': 565.127, 'duration': 6.161}], 'summary': 'Polynomials have finite terms, derivatives are described with an infinite matrix mostly filled with zeros and positive integers, demonstrated using x^3 + 5x^2 + 4x + 5.', 'duration': 37.914, 'max_score': 533.374, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TgKwz5Ikpc8/pics/TgKwz5Ikpc8533374.jpg'}, {'end': 602.208, 'src': 'heatmap', 'start': 572.028, 'weight': 0.706, 'content': [{'end': 574.649, 'text': 'Then put those coordinates on the right of the matrix.', 'start': 572.028, 'duration': 2.621}, {'end': 588.301, 'text': 'the only term that contributes to the first coordinate of the result is one times four, which means the constant term in the result will be four.', 'start': 580.598, 'duration': 7.703}, {'end': 594.324, 'text': 'This corresponds to the fact that the derivative of four x is the constant four.', 'start': 590.543, 'duration': 3.781}, {'end': 602.208, 'text': 'The only term contributing to the second coordinate of the matrix vector product is two times five,', 'start': 595.605, 'duration': 6.603}], 'summary': 'Matrix vector product yields result with constant term of 4 and second coordinate of 10.', 'duration': 30.18, 'max_score': 572.028, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TgKwz5Ikpc8/pics/TgKwz5Ikpc8572028.jpg'}, {'end': 667.665, 'src': 'heatmap', 'start': 623.08, 'weight': 0.965, 'content': [{'end': 624.981, 'text': "And after that, it'll be nothing but zeros.", 'start': 623.08, 'duration': 1.901}, {'end': 634.026, 'text': 'What makes this possible is that the derivative is linear, And, for those of you who like to pause and ponder,', 'start': 626.923, 'duration': 7.103}, {'end': 641.461, 'text': 'you could construct this matrix by taking the derivative of each basis function and putting the coordinates of the results in each column.', 'start': 634.026, 'duration': 7.435}, {'end': 667.665, 'text': 'So surprisingly, matrix vector multiplication and taking a derivative, which at first seemed like completely different animals,', 'start': 660.222, 'duration': 7.443}], 'summary': 'Derivative linearity enables constructing matrix for vector multiplication.', 'duration': 44.585, 'max_score': 623.08, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TgKwz5Ikpc8/pics/TgKwz5Ikpc8623080.jpg'}, {'end': 739.73, 'src': 'embed', 'start': 711.827, 'weight': 4, 'content': [{'end': 715.568, 'text': 'linear transformations and all that stuff should be able to apply.', 'start': 711.827, 'duration': 3.741}, {'end': 722.439, 'text': 'Take a moment to imagine yourself right now as a mathematician developing the theory of linear algebra.', 'start': 717.455, 'duration': 4.984}, {'end': 729.764, 'text': 'You want all of the definitions and discoveries of your work to apply to all of the vectorish things in full generality,', 'start': 723.219, 'duration': 6.545}, {'end': 731.145, 'text': 'not just to one specific case.', 'start': 729.764, 'duration': 1.381}, {'end': 739.73, 'text': 'These sets of vectorish things like arrows or lists of numbers or functions are called vector spaces.', 'start': 733.348, 'duration': 6.382}], 'summary': 'Developing theory of linear algebra for vector spaces.', 'duration': 27.903, 'max_score': 711.827, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TgKwz5Ikpc8/pics/TgKwz5Ikpc8711827.jpg'}, {'end': 796.905, 'src': 'embed', 'start': 770.847, 'weight': 3, 'content': [{'end': 777.873, 'text': "but basically it's just a checklist to make sure that the notions of vector addition and scalar multiplication do the things that you'd expect them to do.", 'start': 770.847, 'duration': 7.026}, {'end': 786.359, 'text': 'These axioms are not so much fundamental rules of nature as they are an interface between you, the mathematician discovering results,', 'start': 779.054, 'duration': 7.305}, {'end': 790.501, 'text': 'and other people who might want to apply those results to new sorts of vector spaces.', 'start': 786.359, 'duration': 4.142}, {'end': 796.905, 'text': 'If, for example, someone defined some crazy type of vector space like the set of all pi creatures,', 'start': 791.442, 'duration': 5.463}], 'summary': 'Checklist ensures expected behavior of vector addition and scalar multiplication for mathematicians and application to new vector spaces.', 'duration': 26.058, 'max_score': 770.847, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TgKwz5Ikpc8/pics/TgKwz5Ikpc8770847.jpg'}, {'end': 949.834, 'src': 'embed', 'start': 919.685, 'weight': 5, 'content': [{'end': 921.706, 'text': 'But as you learn more linear algebra,', 'start': 919.685, 'duration': 2.021}, {'end': 930.15, 'text': 'know that these tools apply much more generally and that this is the underlying reason why textbooks and lectures tend to be phrased well, abstractly.', 'start': 921.706, 'duration': 8.444}, {'end': 936.109, 'text': "So with that, folks, I think I'll call it an end to this Essence of Linear Algebra series.", 'start': 932.168, 'duration': 3.941}, {'end': 943.812, 'text': "If you've watched and understood the videos, I really do believe that you have a solid foundation in the underlying intuitions of linear algebra.", 'start': 936.83, 'duration': 6.982}, {'end': 946.973, 'text': 'This is not the same thing as learning the full topic, of course.', 'start': 944.612, 'duration': 2.361}, {'end': 949.834, 'text': "That's something that can only really come from working through problems.", 'start': 947.133, 'duration': 2.701}], 'summary': 'Essence of linear algebra series provides solid foundation in underlying intuitions.', 'duration': 30.149, 'max_score': 919.685, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TgKwz5Ikpc8/pics/TgKwz5Ikpc8919685.jpg'}], 'start': 416.926, 'title': 'Function spaces and linear algebra', 'summary': 'Covers function spaces, polynomials, basis functions, and their relation to derivative as an infinite matrix, as well as the essence of linear algebra, including vector spaces, axioms, and foundational intuitions for further learning.', 'chapters': [{'end': 669.846, 'start': 416.926, 'title': 'Function spaces and basis functions', 'summary': 'Explains the concept of function spaces using polynomials, defining coordinates and basis functions, and demonstrates the derivative as an infinite matrix with mostly zeros.', 'duration': 252.92, 'highlights': ['The full space of polynomials includes polynomials with arbitrarily large degree, leading to infinitely many coordinates when treated as vectors.', "The basis functions for the polynomials are chosen as pure powers of x, and each individual polynomial's coordinates will be a finite string of numbers with an infinite tail of zeros.", 'The derivative is described with an infinite matrix mostly full of zeros, with the positive integers counting down on the offset diagonal, and the matrix can be constructed by taking the derivatives of each basis function and putting the coordinates of the results in each column.']}, {'end': 986.051, 'start': 671.646, 'title': 'The essence of linear algebra', 'summary': 'Explores the notion of vector spaces in mathematics, emphasizing the abstract nature of vectors, the establishment of axioms, and the general applicability of linear algebra tools, with a focus on providing solid foundational intuitions for further learning.', 'duration': 314.405, 'highlights': ['The chapter emphasizes the abstract nature of vectors and the establishment of axioms for vector spaces.', 'The chapter underscores the general applicability of linear algebra tools to various vectorish things in mathematics.', 'The chapter concludes by emphasizing the importance of solid foundational intuitions for further learning in linear algebra.']}], 'duration': 569.125, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/TgKwz5Ikpc8/pics/TgKwz5Ikpc8416926.jpg', 'highlights': ['The full space of polynomials includes polynomials with arbitrarily large degree, leading to infinitely many coordinates when treated as vectors.', "The basis functions for the polynomials are chosen as pure powers of x, and each individual polynomial's coordinates will be a finite string of numbers with an infinite tail of zeros.", 'The derivative is described with an infinite matrix mostly full of zeros, with the positive integers counting down on the offset diagonal, and the matrix can be constructed by taking the derivatives of each basis function and putting the coordinates of the results in each column.', 'The chapter emphasizes the abstract nature of vectors and the establishment of axioms for vector spaces.', 'The chapter underscores the general applicability of linear algebra tools to various vectorish things in mathematics.', 'The chapter concludes by emphasizing the importance of solid foundational intuitions for further learning in linear algebra.']}], 'highlights': ['Vectors can be perceived as either a list of numbers or a deeper spatial concept, challenging the conventional understanding of vectors as coordinates.', 'Core topics like determinants and eigenvectors in linear algebra are indifferent to the choice of coordinate systems, showcasing the spatial essence of vectors.', 'Functions can be viewed as vectors, with the ability to add and scale, analogous to vector operations.', 'The full space of polynomials includes polynomials with arbitrarily large degree, leading to infinitely many coordinates when treated as vectors.', 'The chapter questions the fundamental nature of vectors and their underlying spatial essence, challenging the conventional perception of vectors as lists of real numbers.', 'The addition of two functions results in a new function f plus g, with output as the sum of values.', 'The application of properties to derivatives shows that the derivative of a sum equals the sum of derivatives.', 'The chapter emphasizes the abstract nature of vectors and the establishment of axioms for vector spaces.', 'Basis vectors are important in completely describing linear transformations, applicable to both arrows and functions.', 'The chapter underscores the general applicability of linear algebra tools to various vectorish things in mathematics.', 'The chapter concludes by emphasizing the importance of solid foundational intuitions for further learning in linear algebra.', 'The derivative is described with an infinite matrix mostly full of zeros, with the positive integers counting down on the offset diagonal, and the matrix can be constructed by taking the derivatives of each basis function and putting the coordinates of the results in each column.', "The basis functions for the polynomials are chosen as pure powers of x, and each individual polynomial's coordinates will be a finite string of numbers with an infinite tail of zeros."]}