4. (45 marks) Let S = {(0,0), (0, 1), (1,0), (1, 1)} CR² and consider the vector space RS. a) (10 marks) Show that if 1 (m, n)-(0,1) fi(m, n) 1 (m, n)- (0,0) 0 (m, n) (0,0) fa(m, n) = (0 (m, n) + (0,1) (m, n)-(1,0) 1 fa(m, n)- = fa(m, n) = (m, n) = (1,1) (1,1) 0 (m, n) (1,0) (m, n) the set {f1, 12, 13, 14) is a basis for Rs. b) (5 marks) Show that (f1, f2, f3, f4) is a frame RS. c) (5 marks) For fERS let Lf(m, n) = f(m, m). Show L is a linear map from RS to RS. d) (10 marks) Write down the matrix that represents L in the frame (f1, f2, f3, f4). e) (5 marks) For f, g € RS let 1 β(f,g) = ΣΣ f(m,n)g(m,n) m=0 n=0 Show that is a bilinear form on RS. f) (10 marks) Write down the matrix that represents in the frame (f1, f2, f3, f4)-