HW 27 extra problems

Let S = {v₁, ..., v_m} and T = {w₁, ..., w_n} be sets of vectors in R^k. Prove or give a counterexample: if every vector in S is orthogonal to every vector in T, then every vector in span(S) is orthogonal to every vector in span(T).

(This problem is helpful for doing problem 29 in Sec 5.2.) Let W be a subspace of Rⁿ and let u and v be vectors in of Rⁿ. Prove proj_W(u + v) = proj_W(u) + proj_W(v).