Linear Algebra Notes

Practice72 MCQs with answers & mark schemes →

A complete, self-contained Linear Algebra course — free, and open to anyone, anywhere. 12 lessons published so far — each with notes, worked examples, a diagram you can move, flashcards and a quiz — and more are being written.

Also called Free online course, Self-study course.

01

Vectors

▸VectorsArrows, lists of numbers, and the two operations that define everything · 22 min · 6 questions→

02

Linear Combinations, Span & Independence

▸Linear Combinations, Span & IndependenceWhich vectors you can reach, and which ones are pulling their weight · 22 min · 6 questions→

03

Matrices

▸MatricesA grid of numbers that packages a whole set of linear combinations · 24 min · 6 questions→

04

Linear Transformations

▸Linear TransformationsA matrix is a function that moves space while keeping the grid straight · 26 min · 6 questions→

05

Systems of Linear Equations

▸Systems of Linear EquationsAx = b, and the elimination that solves it every time · 24 min · 6 questions→

06

Inverse Matrices

▸Inverse MatricesThe matrix that undoes a matrix — and exactly when one exists · 22 min · 6 questions→

07

Determinants

▸DeterminantsOne number that measures how a transformation scales area — and whether it can be undone · 22 min · 6 questions→

08

Vector Spaces & the Four Subspaces

▸Vector Spaces & the Four SubspacesThe spaces a matrix reaches into and the space it kills · 24 min · 6 questions→

09

Orthogonality, Projections & Least Squares

▸Orthogonality, Projections & Least SquaresRight angles, shadows, and the best answer when there is no exact one · 24 min · 6 questions→

10

Eigenvalues & Eigenvectors

▸Eigenvalues & EigenvectorsThe directions a transformation cannot turn — only stretch · 26 min · 6 questions→

11

The Singular Value Decomposition

▸The Singular Value DecompositionEvery matrix is a rotation, a stretch, and another rotation · 24 min · 6 questions→

12

Applications of Linear Algebra

▸Applications of Linear AlgebraWhere all of this quietly runs the modern world · 22 min · 6 questions→

Frequently asked questions

What does Linear Algebra cover?

All 12 chapters — Vectors, Linear Combinations, Span & Independence, Matrices, and more. The finished lessons carry notes, worked examples, an interactive diagram and a quiz; the remaining topics are mapped and being written.

Are there practice questions or past-paper-style questions?

Yes — 72 multiple-choice questions, all with answers, on the Linear Algebra practice page.

Are the Linear Algebra notes free?

Yes — completely free, with no sign-up or login. Every topic has notes, worked examples, an interactive diagram you can change, flashcards and a quiz that explains each answer.