Page 364 - 35Linear Algebra

364 Sample Final Exam

(g) Just use the matrix recursion relation of part (d) repeatedly:

n
2
X n+1 = MX n = M X n−1 = · · · = M X 1 .
√ √
(h) The eigenvalues are ϕ = 1+ 5 and 1 − ϕ = 1− 5 .
2 2
(i)

F n+1 n n −1
X n+1 = = M X n = PD P X 1
F n
n 1 ! 1 !
√ n √
ϕ 0 ? 1 ϕ 0
= P 1 5 = P n 1 5
0 1 − ϕ − √ ? 0 0 (1 − ϕ) − √
5 5
√
√ ! ϕ n ! !
1+ 5 1− 5 √ ?
5
= 2 2 (1−ϕ) n = ϕ −(1−ϕ) n .
n
1 1 − √ √
5 5
Hence
n
ϕ − (1 − ϕ) n
F n = √ .
5
These are the famous Fibonacci numbers.
15. Call the three vectors u, v and w, respectively. Then
1
 
4
u v 3  − 3
⊥
v = v −  4 
 ,
u u u = v − u =  1
4
4 

1
4
and
 
−1
u w v ⊥ w 3 3  0 
⊥
⊥
⊥
w = w − u − v = w − u − 4 v =  
u u v ⊥ v ⊥ 4 3  0 
4
1
Dividing by lengths, an orthonormal basis for span{u, v, w} is
1 3  √ 
   √  
2
 − 2 
 6 
 2 
   √  
  
 1   − 3  0  
√  ,   .
 2  2 
  , 
 0 
 1   3 
  
√
 2  6  √ 
 
1 3
 
 2 
2 6 2
16. (a) The columns of M span imL in the basis given for W.
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