all domains 22 concepts · 4 acts

Linear Algebra

The math of space, motion, and data, seen instead of memorized. Every idea here is a transformation you can picture. This is the exact spine you climb before machine learning, physics, or graphics.

Act I

The objects

6 concepts

1
The vector
Start with one arrow, and the decision that makes everything else possible.
1 explanation
2
Adding and scaling
Only two operations exist here. Everything later is built from them.
1 explanation
needs first: vectors
3
Linear combinations
Scale, add, and find out how much of the plane you can cover.
1 explanation
needs first: vector arithmetic
4
Span and basis
Everything a set of vectors can reach, and the smallest set that reaches it.
1 explanation
needs first: linear combination
5
Independence
A test for whether a new vector is pulling its weight.
1 explanation
needs first: span and basis
6
Subspaces and dimension
The flat worlds living inside a bigger space, and how to count their size.
1 explanation
needs first: linear independence

Act II

The motions

4 concepts

7
Transformations
Stop moving one arrow. Move all of space at once.
1 explanation
needs first: span and basis
8
The matrix
Why that row-times-column rule is not arbitrary at all.
1 explanation
needs first: linear transformations
9
Multiplication
Do one motion, then another, and collapse both into a single matrix.
1 explanation
needs first: the matrix
10
Into three dimensions
One more basis vector, and nothing else changes.
1 explanation
needs first: matrix multiplication

Act III

The structure

5 concepts

11
The determinant
One number for how much a transformation stretches space, and whether it flips.
1 explanation
needs first: linear transformations
12
Determinant as volume
The same number in 3D, measuring volume and catching flatness.
1 explanation
needs first: determinant, transformations 3d
13
Rank and collapse
Two matrices that look almost identical. One loses a whole dimension.
1 explanation
needs first: determinant, subspaces
14
Running it backwards
Solving a system is asking a motion to reverse. Sometimes it cannot.
1 explanation
needs first: determinant, rank and collapse
15
The null space
The directions a transformation destroyed, and why solutions come in families.
1 explanation
needs first: rank and collapse, inverse

Act IV

The payoff

7 concepts

16
The dot product
A number measuring how much two vectors agree, and a perfect test for right angles.
1 explanation
needs first: linear transformations
17
The cross product
Two vectors in 3D, and the vector standing perpendicular to both.
1 explanation
needs first: dot product, transformations 3d
18
Change of basis
The same arrow, described in a different language.
1 explanation
needs first: inverse, span and basis
19
Eigenvectors
Almost every arrow gets knocked off its line. Find the ones that refuse.
1 explanation
needs first: determinant, change of basis
20
Diagonalization
Use the eigenvectors as your axes, and the transformation becomes trivial.
1 explanation
needs first: eigenvectors, change of basis
21
Symmetric matrices
The best-behaved matrices of all, with a guarantee you can rely on.
1 explanation
needs first: diagonalization
22
PCA: the destination
Eigenvectors of real data, and why they are the directions that matter.
1 explanation
needs first: symmetric matrices

Missing a topic?

This roster is a proposal, not a decree. If something belongs in Linear Algebra and is not listed, ask for it. Requests are public, and anyone can pick one up.

Request a topic in Linear Algebra See all requests

Where this leads

Machine Learning

This was never abstract for its own sake. Eigenvectors become PCA. Linear transformations become the layers of a neural network. Every step here was already walking toward here.