(a) Let F : R 3 R 3 be defined by F [x y z] T := [x y x + y 0]T . Show that F is a linear transformation and find (i) a basis of ker F and (ii) a basis of im F.

(b) Let A M22 be a nonzero matrix and let U = {X M22|XA = AX}. Show that dim U 2. (Hint: I and A are in U, check.