Matrix Math

Understand matrices as transformations of space and the trade-offs in computing with them.

Getting Started

Matrices as transformations of space.

A Matrix Is a Transformation

A 2×2 matrix transforms all of space at once. Watch the grid and basis vectors morph, see its columns as their landing spots, and the determinant as area scaling.

Core Operations

Multiplying matrices, and what the product does geometrically.

Multiplying Matrices

Matrix multiplication is composition: apply one transformation, then another. See why order matters and what the product matrix does to space.

Determinant & Inverse

Area scaling, invertibility, and undoing a transformation.

Determinant & Inverse

The determinant is how much a matrix scales area; whether it’s zero decides if the transformation can be undone by an inverse.

Decompositions

Factoring a matrix into simpler pieces — LU, QR, eigen and SVD.

Matrix Decompositions

Factor a matrix into simpler pieces: LU, QR, eigen and SVD. See how rotate·scale·rotate rebuilds any matrix, and what eigenvectors are.

Related categories

Foundations