Question

(4) The Fibonacci sequence $F_1, F_2,...$ is defined by $F_1 = 1$, $F_2 = 1$, and $F_n = F_{n-2} + F_{n-1}$ ($n ge 3$). Define $T in mathcal{L}(mathbb{R}^2)$ by $T(x, y) = (y, x + y)$. (a) Show that $T^n(0, 1) = (F_n, F_{n+1})$ for each $n$. (Use induction.) (b) Find the eigenvalues of $T$. (c) Find a basis of $mathbb{R}^2$ 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 $$F_n = frac{1}{sqrt{5}} left( left(frac{1 + sqrt{5}}{2} ight)^n - left(frac{1 - sqrt{5}}{2} ight)^n ight).$$ (Hint: you'll need to express $(0, 1)$ as a linear combination of your eigenvectors.)

          (4) The Fibonacci sequence $F_1, F_2,...$ is defined by
$F_1 = 1$, $F_2 = 1$, and $F_n = F_{n-2} + F_{n-1}$ ($n ge 3$).
Define $T in mathcal{L}(mathbb{R}^2)$ by $T(x, y) = (y, x + y)$.
(a) Show that $T^n(0, 1) = (F_n, F_{n+1})$ for each $n$. (Use induction.)
(b) Find the eigenvalues of $T$.
(c) Find a basis of $mathbb{R}^2$ 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
$$F_n = frac{1}{sqrt{5}} left( left(frac{1 + sqrt{5}}{2}
ight)^n - left(frac{1 - sqrt{5}}{2}
ight)^n 
ight).$$
(Hint: you'll need to express $(0, 1)$ as a linear combination of your eigenvectors.)

$(4) The Fibonacci sequence F1, F2,... is defined by F1 = 1, F2 = 1, and Fn = Fn-2 + Fn-1 (n ge 3). Define T in mathcalL(mathbbR^2) by T(x, y) = (y, x + y). (a) Show that T^n(0, 1) = (Fn, Fn+1) for each n. (Use induction.) (b) Find the eigenvalues of T. (c) Find a basis of mathbbR^2 consisting of eigenvectors of T, so that the matrix of T with respect to that basis is diagonal. (d) Use your answer to (c) to compute T^n(0, 1) in a different way, and conclude that Fn = frac1sqrt5 left( left(frac1 + sqrt52 ight)^n - left(frac1 - sqrt52 ight)^n ight). (Hint: you'll need to express (0, 1) as a linear combination of your eigenvectors.)$

Added by Gina H.

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