1. (15 pts) Let T : P3 - M22 be a linear transformation defined as T(P(x)) = P(0) p(1) 0 p'(0) for any p(x) EP3 where p'(x) is the derivative of the polynomial. (a) Compute the following 1. T(1 + 2x + 3x2 + 4x3) 2. Find all possible p(a) = do + aix + a2x2 + a3x3 such that T(p(x)) = 1. (b) Find ker(T) and im (T). (c) Is T one-to-one? Is Tonto? Justify your answers.2. (10 pts) Consider the following subspace U = {A ( Mnn | P-AP is diagonal} where PE Mnn is a fixed, invertible matrix. (a) Let P 1 3 . Find a basis for U. (Hint: you may want to do part (b) furst.) (b) For a general invertible P E Mnn, find a basis for U using the following process: 1. Let V C Mnn be the set of diagonal matrices and show that T : U - V is an isomorphism where T(A) = P-1AP 2. Find a basis { D1, D2, ..., Dk} for V (you have to figure out what k should be as well). 3. Map the basis for V back to a basis for U by applying T