5. (14 pts) Consider the operator T E L(R" ) which maps (x1, . . . In) > (Xn, X1, 22 .. . In-1). Observe that Th = 1, where I is the identity transformation. Use this fact to answer the following. (a) Show that 1 is always an eigenvalue of T. What is E(1, T)? (b) Show that -1 is an eigenvalue of T iff n is even. What is E(-1, T)? (c) Show that there are no other real eigenvalues of T (besides 1 and -1). (d) For what n is T diagonalizable? You must justify both why T is diagonalizable when n satisfies some condition, and why T is not diagonalizable (over the reals) when n does not satisfy this condition. (e) Describe the adjoint T*. Is T a normal operator