Consider the linear transformation T : P2(R) P2(R) given by T() = 2f', where f' is the usual derivative of f. (a) (4 points) Let S == {1, x, x2} be the standard basis of P2(R). Find [T(S,S)] the matrix of T with respect to S. (b) (4 points) Find the characteristic polynomial of T. Are all the roots in R? (c) (2 points) Find the eigenvalues of T. (d) (6 points) Find a basis for the eigenspace of T corresponding to each eigenvalue. What are their dimensions? (e) (2 points) Does P2(R) have a basis B of eigenvectors of T? (a) (5 points) Find all n x n diagonalizable matrices that have CT(x) = x^. (b) (5 points) Find all n x n diagonalizable matrices that have CT(x) (x - c)n. =