1. Let V be the real vector space of 3 x 3 symmetric matrices. A typical element of V looks like a d d b e f for some a, b, c, d, e, f E R. Let W be the real vector space of degree 3 polynomials in the indeterminate x. A typical element of W looks like a + ba + cx2 + dx3 for some a, b, c, d E R. Let The : V - W be the linear transformation defined by a d e TK d b = k(d - f) + 2ex + (d + f)x2 + (a+ b+ c)23 for some k E R. (a) (10 points) Let A = {Ell, E22, E33, E12 + E21, E13 + E31, E32 + E23} where 1 0 0 0 0 0 0 0 0 Ell = 0 0 0 , E22 = 010 , E33 = 0 0 0 0 0 0 0 0 0 0 01 0 01 0 0 0 E12 + E21 = 1 0 0 , E13 + E31 = 10 0 0 E32 + E23 = 10 1 0 0 0 1 0 0 01 0 Let B = {1, x, x2, x3}. Note that A is a basis of V and B is a basis of W. Determine [T] the matrix of T with respect to A and B. (b) (10 points) Let C be your favorite basis of W that does not have any degree 2 elements. (Recall for example {1 + x, 1 - x} is a basis of P1(R) that does not have any degree 0 elements.) Determine 4 the matrix of T with respect to A and C. (c) (10 points) Explain how your answers to the previous questions are related. Your explanation should include an explicit relation on [7]4 and X in the form of a single matrix equation. (d) (5 points) For what values of k is T onto