Changing the Coefficient Field

The standard constructions described in section 31.5 for R-modules may be applied to vector spaces. In addition, we may extend or restrict the field of scalars, using the functions described here.

ExtendField(V, L) : ModTupFld, Fld -> ModTupFld, MapHom

Given a K-vector space V, with K a field and L an extension of K, construct the L-vector space U = V tensor _K L. The function returns

(a)

the vector space U; and

(b)

the inclusion homomorphism φ : V -> U.

RestrictField(V, L) : ModTupFld, Fld -> ModTupFld, MapHom

Given a K-vector space V, with K a field and L a subfield of K, construct the L-vector space U consisting of those vectors of V having all of their components lying in the subfield L. The function returns

(a)

the vector space U; and

(b)

the restriction homomorphism φ : V -> U.

VectorSpace(V, F) : ModTupFld, Fld -> ModTupFld, Map

KSpace(V, F) : ModTupFld, Fld -> ModTupFld, Map

KMatrixSpace(V, F) : ModTupFld, Fld -> ModTupFld, Map

KModule(V, F) : ModTupFld, Fld -> ModTupFld, Map

Given an n-dimensional K-vector space V, and a subfield F of a finite field or cyclotomic field K such that K has degree m over F, construct a vector space U of dimension mn over the field F. The function returns

(a)

the vector space U; and

(b)

a mapping φ : V -> U such that a vector (v₁, ..., v_i, ..., v_n) of V is mapped into the vector (u₁₁, ..., u_1n, ..., u_i1, ..., u_in, ..., u_n1, ... u_nn ), where (u_i1, ..., u_in) is the field element v_i written as a vector over the subfield F.