I'm having trouble getting started on this one and I'd really appreciate some hints. This question comes from Macdonald's Linear and Geometric Algebra book that I'm using for self study, problem 2.2.4. 1. The problem statement, all variables and given/known data Let U1 and U2 be subspaces of a vector space V. Let U be the set of all vectors of the form u1 + u2, where u1 is a member of U1 and u2 is a member of U2. Show that U is a subspace of V. 2. Relevant equations U inherits it's operations from V (through U1 and U2) and so we have to show that scalar.mult and vec.add are closed for U. a(bv)=(ab)v av + bv = (a+b)v 3. The attempt at a solution Since U1 and U2 are a subset of V, then adding any u1 to u2 may not be closed wrt either U1 or U2 but I think they should be closed wrt V. Intuitively that seems true but I get stuck trying to be more concrete than that.