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Outline of linear algebra

This is an outline of topics related to linear algebra, the branch of mathematics concerning linear equations and linear maps and their representations in vector spaces and through matrices.

Linear equations

Linear equation

System of linear equations
Determinant
- Minor
- Cauchy–Binet formula
Cramer's rule
Gaussian elimination
Gauss–Jordan elimination
Overcompleteness
Strassen algorithm

Matrices

Matrix

Matrix addition
Matrix multiplication
Basis transformation matrix
Characteristic polynomial
Trace
Eigenvalue, eigenvector and eigenspace
- Cayley–Hamilton theorem
- Spread of a matrix
- Jordan normal form
- Weyr canonical form
Rank
Matrix inversion, invertible matrix
- Pseudoinverse
Adjugate
Transpose
- Dot product
- Symmetric matrix
- Orthogonal matrix
- Skew-symmetric matrix
- Conjugate transpose
  - Unitary matrix
  - Hermitian matrix, Antihermitian matrix
Positive-definite, positive-semidefinite matrix
Pfaffian
Projection
Spectral theorem
Perron–Frobenius theorem
List of matrices
- Diagonal matrix, main diagonal
  - Diagonalizable matrix
- Triangular matrix
- Tridiagonal matrix
- Block matrix
- Sparse matrix
- Hessenberg matrix
- Hessian matrix
- Vandermonde matrix
- Stochastic matrix
- Toeplitz matrix
  - Circulant matrix
- Hankel matrix
- (0,1)-matrix

Matrix decompositions

Matrix decomposition

Cholesky decomposition
LU decomposition
QR decomposition
Polar decomposition
Reducing subspace
Spectral theorem
Singular value decomposition
- Higher-order singular value decomposition
Schur decomposition
- Schur complement
- Haynsworth inertia additivity formula

Relations

Matrix equivalence
Matrix congruence
Matrix similarity
Matrix consimilarity
Row equivalence

Computations

Elementary row operations
Householder transformation
Least squares, linear least squares
Gram–Schmidt process
Woodbury matrix identity

Vector spaces

Vector space

Linear combination
Linear span
Linear independence
Scalar multiplication
Basis
- Change of basis
- Hamel basis
Cyclic decomposition theorem
Dimension theorem for vector spaces
- Hamel dimension
Examples of vector spaces
Linear map
- Shear mapping or Galilean transformation
- Squeeze mapping or Lorentz transformation
Linear subspace
- Row and column spaces
- Column space
- Row space
- Cyclic subspace
- Null space, nullity
- Rank–nullity theorem
- Nullity theorem
Dual space
- Linear function
- Linear functional
Category of vector spaces

Structures

Topological vector space
Normed vector space
Inner product space
- Euclidean space
- Orthogonality
- Orthogonal complement
- Orthogonal projection
- Orthogonal group
Pseudo-Euclidean space
- Null vector
- Indefinite orthogonal group
Orientation (geometry)
- Improper rotation
Symplectic structure

Multilinear algebra

Tensor
- Classical treatment of tensors
- Component-free treatment of tensors
- Gamas's Theorem
Outer product
Tensor algebra
- Exterior algebra
- Symmetric algebra
- Clifford algebra
- Geometric algebra

Topics related to affine spaces

Affine space

Affine transformation
Affine group
Affine geometry
Affine coordinate system
Flat (geometry)
Cartesian coordinate system
Euclidean group
Poincaré group
Galilean group

Projective space

Projective transformation
Projective geometry
Projective linear group
Quadric and conic section