r = 3, s = 2, t = 2 u = 1, v = 4, w = 0

4. (12 marks) In M2x2(C) let U = BEMzx2(C) : By]=[} V = {C E M2x2(C) : tr(C) = 0} (Note : tr(C) is the trace of C, which is the sum of the (1,1) and (2,2) entries of C.) (a) Find a basis for U, and a basis for V (b) Find a basis for Un V (c) With no further calculation, using parts (a) and (b), explain (in one sentence) in why M2x2(C) # VOV? 5. (20 marks) Consider the linear transformation T : P2(R) -+ P2(RR) defined by T(p(x)) = p(r + 1) -p(w)z? (a) Compute T(1), T(x) and T(x?). (b) For = = 1, x, ' and B = 1 + x, 1+ 2', r + 2, find the change of basis matrix =[!]s (c) Find the coordinate vectors [x]= and [r' ]8 (d) Find = = and s [T]a (e) Is T an isomorphism