The Fibonacci sequence F1, F2, . . . is defined by F1 = 1, F2 = 1, and Fn = Fn−2 + Fn−1 (n ≥ 3). Define T ∈ L(R2) by T(x, y) = (y, x + y). (a) Show that T n(0, 1) = (Fn, Fn+1) for each n. (Use induction.) (b) Find the eigenvalues of T. 1 2 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 n(0, 1) in a different way, and conclude that Fn = 1 √5 1 + √5 2 !n − 1 − √5 2 !n! .