Page 364 - 35Linear Algebra
P. 364
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.
364