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Basic properties of linear transformations

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A linear transformation maps 0 to 0.
A linear transformation of a linear combination is the linear combination of the linear transformation
A linear transformation is determined by its action on a basis.
Definition of how the action of a linear transformation on a basis extends to the whole space
The solutions of a homogeneous system are the pre-image (of 0) of a linear transformation.

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Definition of how the action of a linear transformation on a basis extends to the whole space

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A linear transformation is determined by its action on a basis.Close match
A linear transformation is given by a matrix with respect to a given basis.Close match
A linear transformation of a linear combination is the linear combination of the linear transformationClose match
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A linear transformation is given by multiplying by its matrix representation with respect to bases of the spacesClose match
A linear transformation is given by a matrix whose columns are the images of the standard basis vectorsClose match
Definition of matrix representation of a linear transformationClose match
A linear transformation is surjective if and only if the image of a basis is a spanning setClose match
Basic properties of linear transformationsClose match
Definition of matrix representation of a linear transformation with respect to bases of the spacesClose match
Definition of matrix representation of a linear transformation from a vector space to itselfClose match
Any linearly independent set can be expanded to a basis for the (sub)spaceClose match
The sum of linear transformations is a linear transformationClose match
A linear transformation is diagonalizable if there is a basis such that each element is an eigenvector of the transformation.Close match
Definition of sum of linear transformationsClose match
Definition of range of a linear transformationClose match
Equivalence theorem for injective linear transformations: The image of a basis for V is a basis for the range of T.Close match
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Eigenvalues agree and eigenvectors correspond through coordinatesClose match
Subspaces associated to a linear transformationRelated

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