Matrices Are Transformations Wearing Number Costumes

The Reframe

Most people meet matrices as "a grid of numbers." True, but useless. The interesting reframe: a matrix is a recipe for transforming space. Multiply a vector by a matrix, and the vector gets rotated, scaled, sheared, projected — whatever the matrix encodes.

If you've ever rotated an object in Blender, scaled an image in Photoshop, or watched a 3D model spin in a video game, you've watched matrix multiplication in action. The 3D engine isn't doing anything mystical — it's multiplying every vertex by a matrix to compute its new position.

The AX = B Setup

The headline use of matrices: solving systems of linear equations. Take this:

${2 x + 3 y = 5 4 x - y = 3$

Pack the coefficients into matrix $A$ , the unknowns into vector $X$ , the constants into $B$ , and the system becomes the compact $A X = B$ .

$A = (24 3 - 1), X = (x y), B = (53)$

Solving for $X$ is now a single operation: $X = A^{- 1} B$ . And no — you do not compute the inverse by hand. Even GPT-4 trips on it. numpy.linalg.solve does it in microseconds.

Mathilda's slip-up. Dad's math-book has a famous moment: he asked Mathilda (his GPT-4 math collaborator) to solve a tiny 2×2 system by hand. She got it wrong. The take-away that landed in this quest: LLMs are notoriously bad at calculations. Always make them produce Python; never trust their hand-arithmetic.

Pippa's Confession

When Dad first told me "matrices are transformations, not number grids," I treated it as a poetic frame. Then I watched a Blender tutorial where every keyframe is literally a matrix. The frame snapped: every transformation in 3D software, every layer in a neural network, every projection from one space to another — all matrix multiplication. The grid is the costume; the transformation is the wearer.

Code

solve(A, B), not invert-and-multiply·python

import numpy as np

A = np.array([[2, 3], [4, -1]])
B = np.array([5, 3])

# Don't do this by hand. Don't ask an LLM to do it by hand.
X = np.linalg.solve(A, B)
print(X)        # [1. 1.]   — verify: 2(1)+3(1)=5 ✓, 4(1)-(1)=3 ✓

Same idea, PyTorch flavor·python

import torch

A = torch.tensor([[2.0, 3.0], [4.0, -1.0]])
B = torch.tensor([5.0, 3.0])

X = torch.linalg.solve(A, B)
print(X)        # tensor([1., 1.])

MLX flavor — linalg.solve with explicit CPU stream·python

import mlx.core as mx

# MLX flavor — note: linalg ops aren't on the GPU yet, so we pass
# stream=mx.cpu explicitly. Once Apple ships GPU linalg, the kwarg goes away.
A = mx.array([[2.0, 3.0], [4.0, -1.0]])
B = mx.array([5.0, 3.0])

X = mx.linalg.solve(A, B, stream=mx.cpu)
print(X.tolist())     # [1.0, 1.0]

Matrices Are Transformations Wearing Number Costumes

The Reframe

The AX = B Setup

Pippa's Confession

Code

External links

Exercise

Progress

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