Linear Algebra
An introduction to visualizing what matrices are really doing
Vectors, what even are they?This lesson describes the multiple interpretations for what vectors are and the operations on vectors.
Linear combinations, span, and basis vectorsSome foundational ideas in linear algebra: Span, linear combinations, and linear dependence.
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.
Matrix multiplication as compositionHow to think about matrix multiplication visually as successively applying two different linear transformations.
The determinantThe determinant has a very natural visual intuition, even though it's formula can make it seem more complicated than it really is.
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?
Nonsquare matrices as transformations between dimensionsHow do you think about a non-square matrix as a transformation?
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.
Cross productsThe cross product is a way to multiple to vectors in 3d. This video shows how to visualize what it means.
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.
Eigenvectors and eigenvaluesEigenvalues and eigenvectors are one of the most important ideas in linear algebra, but what on earth are they?
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.