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 otimes_K L. The function returns

• the vector space U; and
• the inclusion homomorphism phi : V -> U.

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

• the vector space U; and
• the restriction homomorphism phi : V -> U.

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 K such that K has degree m over F, construct a vector space U of dimension mn over the field F. The function returns

• the vector space U; and
• a mapping phi : V -> U such that a vector (v_1, ..., v_i, ..., v_n) of V is mapped into the vector (u_(11), ..., 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.