Elements of M_n as Homomorphisms

The matrix algebra M_n(S) may also be viewed as the module Hom(S⁽ⁿ⁾, S⁽ⁿ⁾). At present this will not happen automatically so that in order to treat elements of M_n(S) as homomorphisms, it is necessary to explicitly coerce the matrix into Hom(S⁽ⁿ⁾, S⁽ⁿ⁾). However, two fundamental homomorphism-type operators are provided for elements of M_n(S).

Image(a) : AlgMatElt -> ModTup

RowSpace(a) : AlgMatElt -> ModTup

Given an element of M_n(S), return the image of the module S⁽ⁿ⁾ under the homomorphism represented by the matrix a (as an element of S⁽ⁿ⁾).

Kernel(a) : AlgMatElt -> ModTup

NullSpace(a) : AlgMatElt -> ModTup

    Al: MonStgElt                       Default: "Default"

Given an element of M_n(S), return the kernel of the homomorphism represented by the matrix a (as an element of S⁽ⁿ⁾).

RowNullSpace(a) : AlgMatElt -> ModTup

NullspaceOfTranspose(a) : AlgMatVElt -> ModTupRng

Given an element of M_n(S), return the row nullspace of the homomorphism represented by the matrix a (as an element of S⁽ⁿ⁾). This is equal to the kernel of the transpose of a.

Restrict(a, V) : AlgMatElt, ModTupRng -> AlgMatElt

Restrict(a, V) : AlgMatElt, ModTupFld -> AlgMatElt

If V simeq S^(m) is a free submodule of S⁽ⁿ⁾ stable by an element a of M_n(S), returns the restriction of the homomorphism a to the stable submodule V, as an element of M_m(S).