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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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A linear transformation is surjective if and only if the image of a basis is a spanning set

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Equivalence theorem for injective linear transformations: The image of a basis for V is a basis for the range of T.Close match
A linear transformation is surjective if and only if the columns of its matrix span the codomain.Close match
The image of a spanning set spans the rangeClose match
A linear transformation is determined by its action on a basis.Close match
A linear transformation is surjective if and only if the rank equals the dimension of the codomain.Close match
A linear transformation is invertible if and only if it is injective and surjectiveClose match
A linear transformation is given by a matrix with respect to a given basis.Close match
Definition of onto/surjective linear transformationClose match
A linear transformation of a linear combination is the linear combination of the linear transformationClose match
The dimension of the domain of a surjective linear transformation is at least the dimension of the codomain.Close match
The range/image of a linear transformation is a subspace.Close match
A linear transformation is injective on its generalized range space.Close match
A linear transformation is given by a matrix whose columns are the images of the standard basis vectorsClose match
Definition of how the action of a linear transformation on a basis extends to the whole spaceClose match
A linear transformation is given by multiplying by its matrix representation with respect to bases of the spacesClose match
Basic properties of linear transformationsClose match
The composition of surjective linear transformations is surjectiveClose match
The image of a linearly independent set under an injective linear transformation is linearly independent.Close match
A linear transformation is onto if and only if its rank equals the number of rows in any matrix representation.Close match
A linear transformation is diagonalizable if there is a basis such that each element is an eigenvector of the transformation.Close match

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