Machine learning

every neural net layer is matrix multiplication

Training a model means finding the matrix W that minimises loss(W·X − y). Linear algebra is the substrate of every ML library.

Data science

PCA, dimensionality reduction

Principal Component Analysis (PCA) finds the eigenvectors of the covariance matrix — used to compress features into 2D for visualisation.

Computer graphics

3D transformations

Every transformation pipeline (model → world → camera → screen) is a chain of 4×4 matrices, run per vertex per frame.

Search engines

PageRank eigenvector

PageRank, latent-semantic indexing, vector-search (FAISS) all reduce to eigen-decomposition or matrix factorisation.

Quantum mechanics

state vectors

States are vectors in Hilbert space, observables are Hermitian operators (matrices), evolution is a unitary matrix — quantum theory IS linear algebra.

Robotics

kinematics and control

Forward and inverse kinematics solve matrix equations to move a robot arm tip to a target position.