Essence of linear algebra

Kicking off the linear algebra lessons, let's make sure we're all on the same page about how specifically to think about vectors in this context.

The fundamental vector concepts of span, linear combinations, linear dependence, and bases all center on one surprisingly important operation: Scaling several vectors and adding them together.

Matrices can be thought of as transforming space, and understanding how this work is crucial for understanding many other ideas that follow in linear algebra.

Multiplying two matrices represents applying one transformation after another. Many facts about matrix multiplication become much clearer once you digest this fact.

What do 3d linear transformations look like? Having talked about the relationship between matrices and transformations in the last two videos, this one extends those same concepts to three dimensions.

The determinant of a linear transformation measures how much areas/volumes change during the transformation.

How to think about linear systems of equations geometrically. The focus here is on gaining an intuition for the concepts of inverse matrices, column space, rank and null space, but the computation of those constructs is not discussed.

Because people asked, this is a video briefly showing the geometric interpretation of non-square matrices as linear transformations that go between dimensions.

Dot products are a nice geometric tool for understanding projection. But now that we know about linear transformations, we can get a deeper feel for what's going on with the dot product, and the connection between its numerical computation and its geometric interpretation.

This covers the main geometric intuition behind the 2d and 3d cross products.

For anyone who wants to understand the cross product more deeply, this video shows how it relates to a certain linear transformation via duality. This perspective gives a very elegant explanation of why the traditional computation of a dot product corresponds to its geometric interpretation.

This rule seems random to many students, but it has a beautiful reason for being true.

How do you translate back and forth between coordinate systems that use different basis vectors?

A visual understanding of eigenvectors, eigenvalues, and the usefulness of an eigenbasis.

How to write the eigenvalues of a 2x2 matrix just by looking at it.

This is really the reason linear algebra is so powerful.