NapkinCalc

Linear Algebra A — Matrices & Determinants

Matrices as machines

ELI5: a matrix is a grid of numbers that acts on space — feed it a point and it moves that point (rotating, stretching, shearing). Enter one with rows separated by ;.

M:=[2,1;1,3]M := [2, 1; 1, 3] = [[2,1],[1,3]][[2, 1], [1, 3]] a 2×2 machine
Mt=transpose(M)M_{t} = \mathrm{transpose}\left(M\right) = [[2,1],[1,3]][[2, 1], [1, 3]] transpose: flip across the diagonal (rows ↔ columns)

Real-world hook: every time a game rotates a sprite, a phone straightens a photo, or an engineer rotates a stress tensor, a matrix is doing the work.