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.
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.
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.
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, ..., vi, ..., vn) of V is mapped into the vector (u11, ..., u1n, ..., ui1, ..., uin, ..., un1, ... unn ), where (ui1, ..., uin) is the field element vi written as a vector over the subfield F.