u is a finite dimensional vector space . for any subspace S of U , we define S = [ TEN': V X ES , T (X) = 09 if s is a subspace of U then S' is a subspace of u' D . If S, and S, are subspace of U. prove that S, = S. if and only if SP = SZ 3 . If S, and S. are subspaces of U. prove that ( sit s . ) = spas , and ( sins. ) = sit so 3. If S is a subspace of U. prove that dim (U) = dim(s )+lim's)