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Matrices as linear transformations

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Matrices act as a transformation by multiplying vectors
Matrix-vector multiplication is a linear transformation.
Definition of the standard matrix for a linear transformation
A linear transformation is given by a matrix whose columns are the images of the standard basis vectors
A linear transformation is given by a matrix with respect to a given basis.
Definition of nilpotent linear transformation
Definition of matrix representation of a linear transformation
Matrix representation of a composition of linear transformations is given by a matrix product

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Definition of the standard matrix for a linear transformation

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A linear transformation is given by a matrix whose columns are the images of the standard basis vectorsClose match
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Definition of matrix representation of a linear transformation with respect to bases of the spacesClose match
Definition of matrix representation of a linear transformationClose match
Definition of matrix representation of a linear transformation from a vector space to itselfClose match
Matrices act as a transformation by multiplying vectorsClose match
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A linear transformation is given by a matrix with respect to a given basis.Close match
Basic terminology and notationClose match
A linear transformation is given by multiplying by its matrix representation with respect to bases of the spacesClose match
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The set of m by n matrices is a vector space.Close match
Matrix-vector multiplication is a linear transformation.Close match
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The matrix representation of a scalar multiple of linear transformations is the scalar multiple of the matrix.Close match
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Definition of matrix multiplication in terms of column vectorsClose match
Matrix representation of a composition of linear transformations is given by a matrix productClose match
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