Linear Algebra for AI

Linear Algebra for AI — Part 1: What Is Linear Algebra? (The Big Picture)

Last updated: 23 Nov 2025

Linear algebra is the study of straight-line relationships and flat spaces — how things move, stretch, rotate, or combine without breaking structure.

It’s the language of transformations that keep:

1. straight lines → straight
2. parallel lines → parallel
3. the origin → fixed

Think of it as the physics of predictable space.
Everything in AI lives here.

The Fundamental Building Blocks(Everything Stems From These)

Concept	Everyday Meaning
Vector	Arrow with direction + length (or a meaningful list)
Matrix	Grid that transforms vectors — the “warp machine”
Scalar	Simple number that stretches/shrinks vectors
Linear Combination	Mixing scaled vectors (a·v₁ + b·v₂)
Span	All points you can reach with combinations
Basis	Smallest independent set that spans the space

Determinants, eigenvalues, PCA, SVD, neural networks — all built on these six ideas.

Why Does AI Rely So Heavily on Linear Algebra?

AI = vectors pushed through matrices.

1. Data points → vectors in high-dimensional space
2. Neural network layers → matrices that transform those vectors
3. Training → finding the best transformation

No linear algebra → no deep learning.