please explain #17

17. Let T be a linear operator on a finite-dimensional vector space V such that the characteristic polynomial of T splits, and let 1, 12, . .., At be the distinct eigenvalues of T. For each i, let v; denote the unique vector in Kx, such that a = v1 + 12 + ... + v. (This unique representation is guaranteed by Theorem 7.3 (p. 479).) Define a mapping S: V - V by S(x) = 101 + 12 2 + . .. + dRUK. (a) Prove that S is a diagonalizable linear operator on V. (b) Let U = T - S. Prove that U is nilpotent and commutes with S, that is, SU = US