Page 349 - 35Linear Algebra

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well into the future. Then F 3 = 2 because the eggs laid by the first pair of
doves in year two hatch. Notice also that in year three, two pairs of eggs are
laid (by the first and second pair of doves). Thus F 4 = 3.

(a) Compute F 5 and F 6 .
(b) Explain why (for any n ≥ 2) the following recursion relation holds

F n = F n−1 + F n−2 .

F n
(c) Let us introduce a column vector X n = . Compute X 1 and X 2 .
F n−1
Verify that these vectors obey the relationship

1 1
X 2 = MX 1 where M = .
1 0
(d) Show that X n+1 = MX n .
(e) Diagonalize M. (I.e., write M as a product M = PDP −1 where D is
diagonal.)
n
(f) Find a simple expression for M in terms of P, D and P −1 .
n
(g) Show that X n+1 = M X 1 .
(h) The number
√
1 + 5
ϕ =
2
is called the golden ratio. Write the eigenvalues of M in terms of ϕ.
(i) Put your results from parts (c), (f) and (g) together (along with a short
matrix computation) to find the formula for the number of doves F n
in year n expressed in terms of ϕ, 1 − ϕ and n.

15. Use Gram–Schmidt to find an orthonormal basis for

     
 1 1
 0 

1 0 0
 
     
,
span       .
,
 1 1
     

 1 
 
1 1 2
16. Let M be the matrix of a linear transformation L : V → W in given bases
for V and W. Fill in the blanks below with one of the following six vector
⊥ ⊥
spaces: V , W, kerL, kerL , imL, imL .
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