GraphMath

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.

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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.

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Example calculations

Row reduction

A step-by-step example of reduced echelon transformation

Row reduction example showing two elimination steps using the first pivot
Open topic Full solution PDF

Gram-Schmidt

A step-by-step orthogonalization example

Gram-Schmidt example showing construction of the second orthogonal vector by subtracting projection
Open topic Full solution PDF

Visual explanations

Visual linear algebra explanations

Standalone diagrams and animations

Browse visual explanations of selected linear algebra ideas.

Open visual explanations

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 in 2D
2D grid geometry, scaling, shear, rotation, reflection and the bridge to higher-dimensional maps
Linear transformations in 3D
3D scaling and shear matrices acting on the unit cube
Inverse
Definition, existence, uniqueness and computation
Determinant
Scaling factor and structure via row operations
Determinant formulas: permutations and cofactors
Signed permutation terms, cofactors, minors and recursive determinant formulas
Determinant visualization
Orientation, projected cyclic order, recursive projection and cofactor signs in 3D and 4D
Cramer's rule
Solving a square full-rank system from determinant ratios
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
Change of basis
Basis coordinates, inverse basis matrices, coordinate conversion and dimension-lowering change of basis
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
Reflection
Reflection across a subspace by preserving one component and negating its orthogonal complement
Cross product
Direction, magnitude, orientation and the skew-symmetric cross product matrix in ℝ³
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, Gram-Schmidt, part 3
Gradual orthogonalization, stepwise construction of Q and R and right-side shear and scaling updates
Givens rotations
QR factorization by coordinate-plane rotations that eliminate below-diagonal entries
Householder reflection
QR factorization by hyperplane reflections that zero all entries below a pivot at once
QRF & equations
Using QR factorization to solve consistent and inconsistent systems
Best solution & OLS
Best-fit models as projections onto model subspaces
Eigenvectors and eigenvalues
Under construction — geometric meaning, examples and diagonalization

Related external resource

Learn Linear Algebra

External project

A free external project by Prashant Kulkarni exploring linear algebra with GitHub notes, NotebookLM, SageMath/Python, examples and exercises.

Visit project