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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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Definition of leading entry in a row of a matrix#23481
Definition of least-squares error of a linear system#23947
Definition of least-squares solution to a linear system#23946
Definition of left inverse of a matrix#23515
Definition of length/norm of a vector#24090
Definition of linear combination of vectors#23800
Definition of linear dependence relation on a set of vectors#23816
Definition of linear equation#23413
Definition of linearly dependent set of vectors: one of the vectors can be written as a linear combination of the other vectors#23821
Definition of linearly independent set of vectors: if a linear combination is 0#23819
Definition of linear transformation/homomorphism#24013
Definition of LU decomposition#23625
Definition of Markov matrix#23595
Definition of matrix#23449
Definition of matrix diagonalization#23610
Definition of matrix equation#23764
Definition of matrix in reduced row echelon form#23543
Definition of matrix multiplication#23489
Definition of matrix multiplication in terms of column vectors#23488
Definition of matrix null space (left)#23874
Definition of matrix null space (right)#23873
Definition of matrix representation of a linear system#23759
Definition of matrix representation of a linear transformation#23861
Definition of matrix representation of a linear transformation from a vector space to itself#24070
Definition of matrix representation of a linear transformation with respect to bases of the spaces#24068

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