(d).( Hint:Let=[ a b ] and compute Tr(AEij) for each Eij c dbelong to C )

Let & = {e1, e2, ez } be the standard basis of R and let C be the standard basis of M2(R), that is, Let A E M2(R) and consider the linear transformation T : M2(R) R 3 Tr( B) B Tr(AB) Tr( A2 B) Suppose 0 0 ] &-C = 2 3 6 7 14 21 42 (a) (2 points) Compute TS 3 -2 using [T] . Show you work. (Do not use part (d)). (b)(8 points) Find a basis for Ker(T) and a basis for Im(T) . Justify your answer. (c) (2 points) What are rank(T) and nullity (T) ? Justify your answer. (d) (4 points) Find the matrix A. Justify your