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! .

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answer: x=3 and y=4

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guess at question:

a vertical line intersects the graph of g(x,y) = 0 at two points. what conclusion follows about the function f(x) implicitly defined by g(x,f(x)) = 0?

the domain of f(x) cannot include the x coordinate of any such vertical line.

informally, f(x) is not a function, and

g(x,y)=0 cannot be solved for y uniquely.

the relation r = {(x,y) such that g(x,y) = 0} is a function if

for all (a,b) ∊ r and (c,d) ∊ r, a=c implies b=d.

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