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Contents

Vectors and matrices

Vectors

  1. Vectors , essential
  2. Linear combinations and span , essential
  3. Length and the dot product , essential

Matrices as maps

  1. A matrix moves the plane , essential
  2. A gallery of linear maps
  3. Matrix multiplication , essential

Solving linear equations

Linear systems

  1. Two pictures of a linear system , essential
  2. Elimination , essential
  3. The inverse matrix , essential
  4. LU factorisation
  5. Transposes and permutations

Subspaces

  1. Vector spaces and subspaces , essential
  2. The null space , essential
  3. The complete solution
  4. Independence, basis and dimension , essential
  5. The four fundamental subspaces , essential
  6. Graphs and networks

Orthogonality and determinants

Projections and least squares

  1. Projections , essential
  2. Least squares , essential
  3. Orthonormal bases and Gram–Schmidt , essential
  4. QR factorisation
  5. Functions as vectors

Determinants

  1. The determinant as area , essential
  2. Computing determinants

Eigenvalues and eigenvectors

Eigenvalues

  1. Eigenvectors , essential
  2. Finding eigenvalues , essential
  3. Diagonalisation and matrix powers , essential
  4. Markov matrices
  5. Linear differential equations
  6. Complex eigenvalues
  7. Similar matrices and the Jordan form

Symmetric matrices

  1. Symmetric matrices , essential
  2. Positive definite matrices , essential
  3. Complex vectors and the Fourier matrix

The singular value decomposition

The SVD and its uses

  1. The singular value decomposition , essential
  2. Low-rank approximation
  3. Principal components
  4. The pseudo-inverse
  5. Conditioning

Coordinates and transformations

  1. Change of basis , essential
  2. The matrix of a linear map
  3. Matrices in computer graphics

Computing with large matrices

  1. Power iteration and ranking pages
  2. Iterative solvers

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