# Basis & Dimension

Canonical: https://manyspace.aporia.systems/topics/basis-dimension-mth-linalg-159
Updated: 2026-10-13

Mathematics · Linear Algebra · MTH-LINALG-159

- Difficulty: 3 / 5
- Importance: 5 / 5
- Study time: 3–6 h

basis characterization, dimension theorem, coordinate vectors

## Why it matters

Gives every vector space a coordinate system; dimension is the fundamental invariant.

## Where it is used

Compression (PCA/SVD), solving systems, signal representation (Fourier basis).

## How you know you have learned it

Complete 3 independent examples correctly + explain common failure modes.

## What you need first

- [Vector Spaces & Subspaces](https://manyspace.aporia.systems/topics/vector-spaces-subspaces-mth-linalg-158) (required)
- [Polynomial Functions](https://manyspace.aporia.systems/topics/polynomial-functions-mth-precalc-077) (helpful)
- [Direct Proof](https://manyspace.aporia.systems/topics/direct-proof-mth-proofs-104) (helpful)

## What it opens

- [Linear Maps & Matrices](https://manyspace.aporia.systems/topics/linear-maps-matrices-mth-linalg-161)
- [Inner Product Spaces](https://manyspace.aporia.systems/topics/inner-product-spaces-mth-linalg-164)

## Resources

- Linear Algebra Done Right — Sheldon Axler
- Linear Algebra Done Right — Sheldon Axler
- Introduction to Linear Algebra — Gilbert Strang