3Blue1Brown Series Season 1, Episode 9: Dot products and duality

Vectors, what even are they?

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.

Linear combinations, span, and basis vectors

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.

Linear transformations and matrices

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.

Matrix multiplication as composition

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

Three-dimensional linear transformations

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

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

Inverse matrices, column space and null space

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.

Nonsquare matrices as transformations between dimensions

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

Cross products

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

Cross products in the light of linear transformations

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.

Cramer's rule, explained geometrically

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

Change of basis

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

Eigenvectors and eigenvalues

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

A quick trick for computing eigenvalues

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

13 min

Abstract vector spaces

This is really the reason linear algebra is so powerful.

17 min