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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 injective on its generalized range space.
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Linear transformations