For any n X m matrix A = (ajj) in Math,m (R), define its transpose A be the m X n matrix B = (bij) so that bij = aji. (a) Show that the map T : Math,m (R) - Matm, (R); Am At is an injective and surjective linear map. (b) Let A E Math,m (R) and B E Matm,p(R) be an n X m and am X p matrix, respectively. Show (AB)' = B'At. (c) Show for any A E Math,m (R) that (At) = A. (d) Show that if A E Math,n (R) is invertible, then A is also invertible and ( At ) -1 = (A-1)