Let and be subspaces of a vector space V. Let be the set of all vectors v in V such that , where is in and is in . Show that is a subspace of V.
It should be easy to show that is a vector space (please let me know if you need a little help there), from there, we just need to prove that , assume and since and and is a vector space, then it particulary is a group, therefore, addition is closed, and so for any given , and so we have .