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Linear transformations

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Eigenvalues and eigenvectors
Terminology
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
Composition
The preimage of a vector is a translation of the kernel of the linear transformation
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 the image of a spanning set is a spanning set for the range space
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
A linear transformation is diagonalizable if there is a basis such that each element is an eigenvector of the transformation.
Subspaces associated to a linear transformation
The rank plus the nullity of a linear transformation equals the dimension of the domain.
The image of a linearly dependent set under a linear transformation is linearly dependent.
A linear transformation is onto if and only if its rank equals the number of rows in any matrix representation.
A linear transformation is invertible if and only if it is injective and surjective
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 representation of a sum of linear transformations is the sum of the matrices.
The matrix representation of a composition of linear transformations is the product of the matrices.
The matrix representation of the inverse of linear transformations is the inverse of the matricix.
A linear transformation has the same eigenvalues and eigenvectors as any matrix representation.
A linear transformation has a representation as an upper triangular matrix.
Equivalence theorems for injective transformations

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A linear transformation is diagonalizable if there is a basis such that each element is an eigenvector of the transformation.

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A set of vectors is linearly independent if and only if the matrix of column vectors in reduced row-echelon form has every column as a pivot column.#23824
A subset of a linearly independent set is linearly independent.#23980
A vector can be written uniquely as a linear combination of vectors from independent subspaces.#23973
A vector can be written uniquely as a sum of a vector in a subspace and a vector orthogonal to the subspace.#23934
A vector is in the orthogonal complement of a subspace if and only if it is orthogonal to every vector in a basis of the subspace.#23923
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
Block matrices#23567
Canonical forms of matrices#23608
Change of coordinates matrices are invertible#23998
Characteristic and minimal polynomials#23699
C^n is a vector space.#24098
Cofactors#23716
Composition#24022
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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