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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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A linear transformation is given by a matrix whose columns are the images of the standard basis vectors

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Definition of the standard matrix for a linear transformationClose match
Matrices as linear transformationsClose match
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 with respect to a given basis.Close match
Matrices act as a transformation by multiplying vectorsClose match
A linear transformation is determined by its action on a basis.Close match
Matrix-vector multiplication is a linear transformation.Close 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
Definition of matrix representation of a linear transformation with respect to bases of the spacesClose match
The matrix representation of a scalar multiple of linear transformations is the scalar multiple of the matrix.Close match
Example of a linear transformation on R^2: genericClose match
A linear transformation of a linear combination is the linear combination of the linear transformationClose match
The matrix representation of a sum of linear transformations is the sum of the matrices.Close match
Definition of how the action of a linear transformation on a basis extends to the whole spaceClose match
Equivalence theorem for injective linear transformations: The columns of the matrix of T are linearly independent.Close match
A linear system is equivalent to a vector equation.Close match
A linear transformation is diagonalizable if there is a basis such that each element is an eigenvector of the transformation.Close match
Visualise a linear transformation on R^2 by looking at the image of a regionClose match
ExamplesClose match

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