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Linear transformations

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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
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

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Definition of matrix null space (right)
Definition of matrix null space (left)
Definition of column space of a matrix
The column space of an m-by-n matrix is a subspace of R^m (or C^m)
The pivot columns of a matrix are a basis for the column space.
Row operations do not necessarily preserve the column space.
Nonzero rows of reduced row-echelon form give a basis for the row space
Definition of row space of a matrix
Row equivalent matrices have the same row space.

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Subspaces associated to a matrix

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The left null space of a matrix is a subspace of R^m (or C^m).Close match
Definition of matrix null space (left)Close match
The null space of a matrix is a subspace of R^n (or C^n).Close match
Definition of matrix null space (right)Close match
The row space of a matrix is a vector spaceClose match
The column space of an m-by-n matrix is a subspace of R^m (or C^m)Close match
Null spaces are orthogonal complements of the corresponding row or column spacesClose match
The column space of a matrix is a vector spaceClose match
Definition of row space of a matrixClose match
Rank and nullityClose match
Theorem describing the dimension of spaces associated to the block matrices of the extended reduced row echelon form of a matrixClose match
Definition of column space of a matrixClose match
Nonzero rows of reduced row-echelon form give a basis for the row spaceClose match
Row operations do not necessarily preserve the column space.Close match
Row equivalent matrices have the same row space.Close match
Rank–nullity theorem for an m-by-n matrixClose match
Matrices as linear transformationsClose match
Subspaces associated to a linear transformationClose match
Description of a basis for the null space of a matrix from the reduced row-echelon form.Close match
An eigenspace of a matrix is the null space of a related matrix.Close match

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