Let [n] = {1, 2, ..., n}. Suppose Al, . ..; Am are sets such that m UAi = [n], i=1 each |A | has even size and each intersection A; n A, has even size. In this problem we will prove that m F as (v, w) = vTw. (i) Which properties of inner product does (, ) satisfy, and which ones does it not? Let P([n]) be the power set of [n]. We have a map F : P([n]) - V defined as: F(X) = G. iEx (ii) Prove that F is a bijection. iii) Prove that X has even size if and only if | |F(X)I| = 0. (iv) For X, Y C [n], prove that X n Y has even size if and only if (F(X), F(Y) ) = 0. (v) Let W = Span(F(Al), F(A2), . .., F(Am)). Prove that W C W-. (vi) Prove that dim(W) + dim(W !) = dim(V). Hint: Consider the m x n matrix whose rows are F(A,), ..., F(Am)". (vii) Prove that 2m