I have a couple questions that I cannot figure out! My professor did not show up for his office hour. So I REALLY REALLY hope someone can help me. even one question would make my day sooo much better! THANKS~!!

(1) Let T be a normal operator on a finite-dimensional inner product space V.
Prove that N(T) = N(T*) and R(T) = R(T*).

(2) Let T be a normal operator on a finite dimensional real inner product space V whose characteristic polynomial splits. Prove that V has an orthonormal basis of eigenvectors of T. Hence prove that T is self-adjoint.