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Definition of composition of linear transformations
The composition of linear transformations is a linear transformation
The composition of injective linear transformations is injective
The composition of surjective linear transformations is surjective
The composition of invertible linear transformations is invertible
The inverse of a linear transformation is a linear transformation
The inverse of the inverse of a linear transformation is the original linear transformation
The inverse of a composition reverses the order

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The inverse of a composition reverses the order

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The composition of invertible linear transformations is invertibleClose match
The inverse of the inverse of a linear transformation is the original linear transformationClose match
Matrix transpose commutes with matrix inverse.Close match
The inverse of a linear transformation is a linear transformationClose match
Matrix inverses are unique: if A and B are square matricesClose match
The inverse of a scalar multiple is the reciprocal times the inverse.Close match
CompositionClose match
For n-by-n invertible matrices A and BClose match
Matrix inverse is an involution.Close match
Transpose commutes with scalar multiplication.Close match
The conjugate of the transpose is the transpose of the conjugate.Close match
InverseClose match
The composition of injective linear transformations is injectiveClose match
The composition of linear transformations is a linear transformationClose match
The matrix representation of a composition of linear transformations is the product of the matrices.Close match
The composition of surjective linear transformations is surjectiveRelated
Definition of composition of linear transformationsRelated
A linear transformation is invertible if and only if it is injective and surjectiveRelated
Matrix representation of a composition of linear transformations is given by a matrix productRelated
Determinants multiply: det(AB) = det(A)det(B)Related

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