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Linear Algebra

Matrices, determinants, inverses, row reduction and eigenvalues.

Lessons

Core Determinants 2×2 by ad − bc, larger by cofactor expansion, and what the number means. det [[1,2],[3,4]] Core Matrix inverse The 2×2 formula and Gauss-Jordan elimination for the rest. inverse of [[1,2],[3,4]] Advanced Eigenvalues and eigenvectors The characteristic polynomial and the directions a matrix only stretches. eigenvalues of [[2,1],[1,2]] Introductory Matrix multiplication Rows times columns, entry by entry. [[1,2],[3,4]] * [[5,6],[7,8]] Core Row reduction Gaussian elimination, rank and solving Ax = b. rref [[1,2,3],[4,5,6],[7,8,9]] Core Vector spaces, span and linear independence The axioms, spanning sets, independence, basis and dimension. rank of [[1,2],[2,4]] Core Orthogonality, projections and least squares Dot products, orthonormal bases, projection onto a subspace, fitting a line. inverse of [[2,1],[1,1]] Core Diagonalisation and matrix powers A = PDP⁻¹, computing Aⁿ, and the matrices that cannot be diagonalised. eigenvalues of [[4,1],[2,3]] Core Linear transformations and change of basis Matrices as maps, kernel and image, and the same map in different coordinates. [[0,-1],[1,0]] * [[1],[0]]

Vectors and the matrices that move them. Every operation is shown row by row, and 2×2 and 3×3 matrices are drawn in 3D so you can see what a determinant or an eigenvector actually is.

More lessons

Systems of Linear Equations

What is Linear Algebra?Solving Systems of Linear Equations Reduced Row-Echelon Form Types of Solution Sets Homogeneous Systems of Equations Nonsingular Matrices

Vectors

Vector Operations Linear Combinations Spanning Sets Linear Dependence and Spans

Matrices

Matrix Operations Matrix Inverses and Systems of Linear Equations Matrix Inverses and Nonsingular Matrices Column and Row Spaces Four Subsets Matrix (mathematics)Determinant Invertible matrix Transpose Row echelon form Gaussian elimination Trace (linear algebra)Block matrix Elementary matrix

Vector Spaces

Vector Spaces Subspaces Linear Independence and Spanning Sets Bases Dimension Properties of Dimension

Determinants

Determinant of a Matrix Properties of Determinants of Matrices

Eigenvalues

Properties of Eigenvalues and Eigenvectors Similarity and Diagonalization

Linear Transformations

Injective Linear Transformations Surjective Linear Transformations Invertible Linear Transformations

Representations

Vector Representations Matrix Representations Orthonormal Diagonalization

Preliminaries

Complex Number Operations Sets

Systems of Equations and Inequalities

Introduction to Systems of Equations and Inequalities Systems of Linear Equations: Two Variables Systems of Linear Equations: Three Variables Systems of Nonlinear Equations and Inequalities: Two Variables Partial Fractions Matrices and Matrix Operations Solving Systems with Gaussian Elimination Solving Systems with Inverses Solving Systems with Cramer's Rule

Structure

Linear algebra Vector space Linear independence Basis (linear algebra)Dimension (vector space)Linear subspace Span (linear algebra)Linear map Kernel (linear algebra)Rank–nullity theorem Change of basis

Geometry and spectra

Eigenvalues and eigenvectors Characteristic polynomial Diagonalizable matrix Jordan normal form Cayley–Hamilton theorem Inner product space Orthogonality Gram–Schmidt process Orthogonal matrix Symmetric matrix Positive-definite matrix Singular value decomposition Least squares Quadratic form

Some of these lessons are adapted from open texts: Beezer, A First Course in Linear Algebra, OpenStax Precalculus 2e, Wikipedia. Each lesson page names its source and licence.

Symbols used here

Tap any symbol for the full definition, a picture, and what every letter in it means.

Questions people ask

What does a determinant mean geometrically?

It is the factor by which the matrix scales area (2×2) or volume (3×3), with a negative sign if orientation flips. Zero means the matrix flattens space and cannot be undone.

What is an eigenvector?

A direction the matrix does not turn: it only stretches it by the eigenvalue. Along eigenvectors a complicated matrix acts like multiplication by a number.

Why is matrix multiplication not commutative?

Because a matrix is a transformation and AB means "do B, then A". Rotating then reflecting is not the same as reflecting then rotating.

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