# Linear Algebra

An introduction to visualizing what matrices are really doing

Essence of linear algebra previewThe introduction to a series on visualizing core ideas of linear algebra.Chapter 0Aug 5, 2016

Vectors, what even are they?This lesson describes the multiple interpretations for what vectors are and the operations on vectors.Chapter 1Aug 6, 2016

Linear combinations, span, and basis vectorsSome foundational ideas in linear algebra: Span, linear combinations, and linear dependence.Chapter 2Aug 6, 2016

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

Matrix multiplication as compositionHow to think about matrix multiplication visually as successively applying two different linear transformations.Chapter 4Aug 8, 2016

Three-dimensional linear transformationsHow to think of 3x3 matrices as transforming 3d spaceChapter 5Aug 9, 2016

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

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

Nonsquare matrices as transformations between dimensionsHow do you think about a non-square matrix as a transformation?Chapter 8Aug 16, 2016

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

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

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

Cramer's rule, explained geometricallyWhat Cramer's rule is, and a geometric reason it's trueChapter 12Mar 17, 2019

Change of basisWhat is a change of basis, and how do you do it?Chapter 13Sep 11, 2016

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

A quick trick for computing eigenvaluesA quick way to compute eigenvalues of a 2x2 matrixChapter 15May 7, 2021

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