Linear Algebra
What survives a change of viewpoint?
Matrices are usually taught as arrays with rules, which is like teaching music as fingerings. A linear map exists before any coordinates are chosen, and almost everything in the subject is the question of which facts about it depend on that choice and which do not.
Vector spaces
What is a vector when it is not an arrow?
Arrows, polynomials, and functions in one definition
Three collections with nothing in common satisfy the same eight axioms. The film checks all three before the definition is stated.
A basis is a choice, and the number is not
Any two bases of a space have the same size, which is far from obvious and is the reason dimension means anything.
Linear maps
How much can a matrix hide?
The map comes first, the matrix second
The same map has infinitely many matrices, one for each basis. Watching the entries change while the map does not is the point of the film.
What the determinant measures
The determinant is presented as a formula and is actually a signed volume. Once it is a volume, every one of its properties becomes obvious.
Eigenvalues
Which directions does a map leave alone?
Directions that only stretch
Most vectors get turned by a map. A few only get longer, and those few determine nearly everything about it.
When can a map be made diagonal?
Not always — and the obstruction is a genuine mathematical phenomenon rather than a technicality, which the film demonstrates with a single stubborn matrix.
Inner product spaces
What does perpendicular mean without a picture?
Linear Algebra, as one volume
Every document in this subject is a chapter of the same book, compiled from one source with live cross-references and continuous numbering. Download the whole thing, or take chapters as you go — the page numbers and theorem references agree either way.
Download Volume VIII ↓