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Linear transformations
Matrices

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Basic terminology and notation
Operations on matrices
Particular types of matrices
Matrix equivalence
Canonical forms of matrices
Factorization of matrices
Similarity of matrices
Nonsingular matrices and equivalences
Eigenvalues and eigenvectors
The matrix representation of a sum of linear transformations is the sum of the matrices.
A linear transformation is invertible if and only if it is injective and surjective
Subspaces associated to a linear transformation
A linear transformation is diagonalizable if there is a basis such that each element is an eigenvector of the transformation.
The preimage of a vector is a translation of the kernel of the linear transformation
The matrix representation of a composition of linear transformations is the product of the matrices.
Geometric properties of linear transformations
Matrices as linear transformations
Basic properties of linear transformations
Description of a spanning set for the null space of a matrix from the reduced row-echelon form.
Description of a basis for the null space of a matrix from the reduced row-echelon form.
The nonzero rows of an echelon form of a matrix are linearly independent.
Subspaces associated to a matrix
Rank and nullity
Examples
Terminology
Determinants
Composition
The image of a linearly independent set under an injective linear transformation is linearly independent.
The dimension of the domain of an injective linear transformation is at most the dimension of the codomain.
The dimension of the domain of a surjective linear transformation is at least the dimension of the codomain.
A linear transformation is surjective if and only if the rank equals the dimension of the codomain.
The range of a linear transformation is a subspace
The image of a spanning set spans the range
A linear transformation is surjective if and only if the image of a basis is a spanning set
Definition of generalized range space of a linear transformation
A linear transformation is injective on its generalized range space.
Definition of diagonalizable linear transformation
The rank plus the nullity of a linear transformation equals the dimension of the domain.
Linear maps preserve dependence of lists, with image-set caveats
A linear transformation is onto if and only if its rank equals the number of rows in any matrix representation.
Definition of matrix representation of a linear transformation with respect to bases of the spaces
A linear transformation is given by multiplying by its matrix representation with respect to bases of the spaces
Definition of matrix representation of a linear transformation from a vector space to itself
The matrix representation of a scalar multiple of linear transformations is the scalar multiple of the matrix.
The matrix of an inverse is the inverse of the matrix representation
Eigenvalues agree and eigenvectors correspond through coordinates
Triangularization requires the characteristic polynomial to split
Equivalence theorems for injective transformations
Choosing factorizations and understanding numerical error
Rank and nullity

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Particular types of matrices
Definition of eigenvalue of a matrix
Definition of eigenvector of a matrix
Eigenspaces
Every matrix has an eigenvalue over the complex numbers.
Eigenvalues and operations on matrices
Eigenvectors with distinct eigenvalues are linearly independent.
Multiplicity
Characteristic and minimal polynomials
The dimension of a eigenspace is less than or equal to the (algebraic) multiplicity of the eigenvalue.
Definition of eigenvalue/characteristic value of a linear transformation
Definition of eigenvector/characteristic vector of a linear transformation
Definition of characteristic polynomial of a linear transformation
Definition of minimal polynomial of a linear transformation
The Cayley-Hamilton theorem for a linear transformation
The minimal polynomial of a linear transformation exists and is unique.
Definition of applying a polynomial to a linear transformation
Every nonzero finite-dimensional complex operator has an eigenvalue
Definition of eigenspace of a linear transformation
The eigenspace of a linear transformation is a nontrivial subspace.
Definition of invariant subspace of a linear transformation.
Invariant direct sums give block diagonal operator matrices

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Axioms of a vector space#23798
Bases#23828
Basic properties#23957
Basic properties of linear transformations#23863
Basic terminology#23412
Basic terminology and notation#23448
Basis exchange using a nonzero coordinate#23990
Block matrices#23567
Canonical forms of matrices#23608
Change of coordinates matrices are invertible#23998
Characteristic and minimal polynomials#23699
Choosing factorizations and understanding numerical error#37115
C^n is a vector space.#24098
Cofactors#23716
Commuting diagonalizable and nilpotent decomposition over the complex numbers#23585
Complex inner products conjugate scalars in the first argument#23900
Composition#24022
Condition number measures relative sensitivity#23750
Conjugating by a change of coordinates matrix converts matrix representations with respect to different bases.#23999
Conjugation#23464
Container for Linear Algebra#24123
Coordinates#23989
Coordinate vector spaces#23776
Cramer's rule#23722
Definition and terminology#23954

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