Essence of linear algebra
16 episodes · 2016
A geometric understanding of matrices, determinants, eigen-stuffs and more.
1Vectors, 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.
2Linear 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.
3Linear 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.
4Matrix 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.
5Three-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.
6The determinant
The determinant of a linear transformation measures how much areas/volumes change during the transformation.
7Inverse 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.
8Nonsquare 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.
9Dot products and duality
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.
10Cross products
This covers the main geometric intuition behind the 2d and 3d cross products.
11Cross 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.
12Cramer's rule, explained geometrically
This rule seems random to many students, but it has a beautiful reason for being true.
13Change of basis
How do you translate back and forth between coordinate systems that use different basis vectors?
14Eigenvectors and eigenvalues
A visual understanding of eigenvectors, eigenvalues, and the usefulness of an eigenbasis.
15A quick trick for computing eigenvalues
How to write the eigenvalues of a 2x2 matrix just by looking at it.
16Abstract vector spacesfinale
This is really the reason linear algebra is so powerful.
