Let W be a subspace of a vector space V over a field F. For any V belongs to V, the set {v} + W = {v + w|w belongs to W} is called the coset of W containing V. It is customary to denote this coset by V + W rather than {v} + W. Prove that V + W is a subspace of V if and only if V belongs to W. Show that if S_1 and S_2 are subsets of a vector space V such that S_1 subset S_2, then spsa(S_1) subset span(S_2). S_1 = {v_1, v_2...v_n}. Let U, V, and W be distinct vectors of a vector space V. Show that if {u, V, w} is a basis for V, then {u + v + w, v+w, w} is also a basis for V.