9 of 9 written

Linear Algebra

The language underneath everything else: linear maps, from vectors and elimination up to eigenvalues, the SVD, and numerical conditioning.

Part 0

A matrix is a linear map, and its columns are where the basis vectors land, with a grid you can warp.

Part 1

Span, linear independence, basis, and dimension, with two vectors whose span you can collapse to a line.

Part 2

Ax as a combination of columns, rank, and the four fundamental subspaces.

Part 3

Row reduction to upper triangular, pivots, and the A = LU factorization, stepped through with real numbers.

Part 4

Projection, the normal equations, and QR, with a draggable point projected onto a line.

Part 5

Directions a map only scales, the characteristic equation, and diagonalization, shown live.

Part 6

A = UΣVᵀ as rotate, stretch, rotate: the unit circle becomes an ellipse whose axes are the singular values.

Part 7

Orthonormal eigenvectors, quadratic forms, and when xᵀAx is a bowl versus a saddle.

Part 8

The condition number, error amplification, and the iterative solvers behind large-scale simulation.