Linear algebra · 03 · Verbs for space · 9 min
Matrices as transformations
A matrix looks like a grid of numbers. It is — but the grid is a disguise. A matrix is an action: a machine that moves every point in space at once. Rotate, stretch, flip, shear — each is a matrix; applying it is multiplication.
Build the intuition
Read the columns: where do the axes land?
A 2×2 matrix's first column says where the x-arrow (1,0) lands; the second, where the y-arrow (0,1) lands. That's the entire decoding manual. The matrix [[0,−1],[1,0]] sends (1,0)→(0,1) and (0,1)→(−1,0): a quarter turn. Read columns, see the verb.
Composition: verbs chain
Rotate, then stretch? Multiply the two matrices once, and the product is a single matrix that does both. A game engine composes camera, object, and perspective transforms into one matrix per frame — then applies it to millions of points cheaply. Chaining is the superpower.
The determinant: the area receipt
Each transformation scales areas by a fixed factor — the determinant. Det 2: areas double. Det 1: areas preserved (pure rotation). Det 0: space is flattened onto a line, information destroyed, no undo. The determinant is the one-number health report of a matrix.
See it move
Six verbs, one grid. Watch what each matrix does to space itself — and read its columns to see why.
A worked example
Decode a matrix by its columns
What does M = [[2, 0], [0, 1]] do?
Column 1: (1,0) lands on (2,0) — the x-direction doubles. Column 2: (0,1) stays put.
Verdict: a horizontal stretch ×2. Check the determinant: 2·1 − 0·0 = 2 — areas double, consistent.
No formula sheet needed: columns tell the story every time.
Out in the world
Neural networks are matrix stacks
Each layer of a neural network multiplies its input vector by a learned matrix (then bends the result slightly). “Training” means nudging those matrix entries until the chain of transformations turns pixels into “cat.” GPT-class models are mostly very large matrices, multiplied very fast.
Common confusion, cleared
“Matrix multiplication should work like number multiplication.”
It composes actions, so order matters: rotate-then-stretch ≠ stretch-then-rotate. AB ≠ BA isn't a defect — it's faithfulness to how actions combine.
“A matrix is just a table for storing numbers.”
Storage is incidental. The mathematical content is the transformation it performs — the same matrix is a verb, not a spreadsheet.
Recap
- A matrix moves all of space; its columns show where the axes land.
- Multiplying matrices chains transformations into one.
- The determinant reports the area-scaling factor; zero means collapse.