Linear Algebra
An introduction to visualizing what matrices are really doing
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Vectors, what even are they?This lesson describes the multiple interpretations for what vectors are and the operations on vectors.Chapter 1Aug 6, 2016
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Linear combinations, span, and basis vectorsSome foundational ideas in linear algebra: Span, linear combinations, and linear dependence.Chapter 2Aug 6, 2016
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Linear transformations and matricesWhen you think of matrices as transforming space, rather than as grids of numbers, so much of linear algebra starts to make sense.Chapter 3Aug 7, 2016
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Matrix multiplication as compositionHow to think about matrix multiplication visually as successively applying two different linear transformations.Chapter 4Aug 8, 2016
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Three-dimensional linear transformationsHow to think of 3x3 matrices as transforming 3d spaceChapter 5Aug 9, 2016
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The determinantThe determinant has a very natural visual intuition, even though it's formula can make it seem more complicated than it really is.Chapter 6Aug 10, 2016
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Inverse matrices, column space and null spaceHow do you think about the column space and null space of a matrix visually? How do you think about the inverse of a matrix?Chapter 7Aug 15, 2016
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Nonsquare matrices as transformations between dimensionsHow do you think about a non-square matrix as a transformation?Chapter 8Aug 16, 2016
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Dot products and dualityWhat is the dot product? What does it represent? Why does it have the formula that it does? All this is explained visually.Chapter 9Aug 24, 2016
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Cross productsThe cross product is a way to multiple to vectors in 3d. This video shows how to visualize what it means.Chapter 10Sep 1, 2016
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Cross products in the light of linear transformationsThe formula for the cross product can feel like a mystery, or some kind of crazy coincidence. But it isn't. There is a fundamental connection between the cross product and determinants.Chapter 11Sep 1, 2016
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Cramer's rule, explained geometricallyWhat Cramer's rule is, and a geometric reason it's trueChapter 12Mar 17, 2019
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Change of basisWhat is a change of basis, and how do you do it?Chapter 13Sep 11, 2016
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Eigenvectors and eigenvaluesEigenvalues and eigenvectors are one of the most important ideas in linear algebra, but what on earth are they?Chapter 14Sep 15, 2016
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A quick trick for computing eigenvaluesA quick way to compute eigenvalues of a 2x2 matrixChapter 15May 7, 2021
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Abstract vector spacesWhat is a vector space? Even though they are initial taught in the context of arrows in space, or with vectors being lists of numbers, the idea is much more general and far-reaching.Chapter 16Sep 24, 2016