Linear Algebra

Tutorial chapters

Chapters are structured to build understanding step by step, minimizing logic gaps and emphasizing geometric insight.

Browse Chapters Get Linear Algebra World Back to GraphMath

Further references

Reviews & Mentions External Presence

Apps

Linear Algebra World

Main learning app

The main GraphMath app for linear algebra learning, combining structured chapters, visual intuition and interactive tools.

Get Linear Algebra World

Available on iPhone, iPad and Mac.

Matrix Solver Step by Step

Companion solver

A focused computational companion for row reduction, linear systems and matrix calculations with step-by-step output.

Get Matrix Solver

Example calculations

Row reduction

A step-by-step example of reduced echelon transformation

Open topic Full solution PDF

Gram-Schmidt

A step-by-step orthogonalization example

Open topic Full solution PDF

Chapters

First steps

Space, vectors, matrices and the first key ideas

Matrix multiplication

Meaning, computation and interpretation

More on matrix multiplication

More viewpoints, structure and examples

Row reduction

Echelon form, pivots, rank and what the algorithm reveals

Linear equations

Solve A x = b and interpret solutions structurally

Linear transformations

From matrices to geometry and back

Inverse

Definition, existence, uniqueness and computation

Determinant

Scaling factor and structure via row operations

Schur complement

Block matrices and elimination in matrix form

Row view of a matrix

Rows as constraints and their geometric meaning

The four subspaces

Row space, column space, null space and left null space

Inverse & Pseudoinverse

Inverse, left inverse, right inverse and pseudoinverse across square, tall, wide and rank-deficient cases

Rank

Definition, calculation, rank(AB), rank(A+B) and geometric interpretation

Shape & rank

Matrix shape, rank and what is / isn’t solvable

Projection

Orthogonal projection from one direction to a subspace

3D rotation

Projection, Rodrigues algorithm and the structure of rotation around an axis

QRF, Gram-Schmidt, part 1

Gram-Schmidt orthogonalization, the structure of Q and the projection matrix QQᵀ

QRF, Gram-Schmidt, part 2

The structure of R, geometric views of QR factorization and uniqueness of thin QR

QRF & equations

Using QR factorization to solve consistent and inconsistent systems

Best solution & OLS

Best-fit models as projections onto model subspaces