Could you help me on this hw? Thank you.

Prob 1. Let v be an eigenvector of T L(V ) with eigenvalue . Show that (T 3 + T 2 2T + I)v = (3 + 2 2 + 1)v. How does this observation generalize? Prob 2. Let S, T L(V ) and suppose S is invertible. Prove that, for any polynomial p P(IF), p(ST S 1 ) = S p(T ) S 1 . Prob 3. Given an example of an operator T P2 (C) with eigenvalues 3 and 4 which is not diagonalizable. Prob 4. Let V be a nite-dimensional real vector space and let T L(V ). Dene f : IR IR by f () := dim range (T I). Which condition on T is equivalent to f being a continuous function? Prob 5. Suppose V is a nite-dimensional complex vector space, T L(V ) is diagonalizable, and all eigenvalues of T are strictly below 1 in absolute value. Given V and v V , what is limn (T n v)? Prob 6. Explain how diagonalizability of (square) matrices you learned in Math 54 is related to the diagonalizability of operators.