could you help me to solve this problem with a proof, a counterexample, or a detailed computation. with explanation and detail.

(4) The Fibonacci sequence F1, F2, .. . is defined by F1 =1, F2 =1, and Fn = Fn-2 + Fn-1 (n >3). Define TE C(R?) by T(x, y) = (y, x ty). (a) Show that T" (0, 1) = (Fn, Fn+1) for each n. (Use induction.) (b) Find the eigenvalues of T. MATH 436 HOMEWORK 9 DUE FRIDAY APRIL 3 (c) Find a basis of R2 consisting of eigenvectors of T, so that the matrix of T with respect to that basis is diagonal. (d) Use your answer to (c) to compute T (0, 1) in a different way, and conclude that Fn = 1 1 + v5 V5 2 (Hint: you'll need to express (0, 1) as a linear combination of your eigenvectors.)