For any integer m, we define the Fibonacci number
f
m
recursively by
f
0
=
0
,
f
1
=
1
, and
f
j
+
2
=
f
j
+
f
j
+
1
for all integers
j
.
7
a. Find the Fibonacci numbers
f
m
for
m
=
−
5
,
...
,
5
.
b. Based upon your answer in part (a), describe the relationship between
f
−
m
and
f
m
. (For extra credit, give a formal proof by induction on m.)
Now let n be a positive integer, with
n
≥
2
. Let V be the two-dimensional subspace of all vectors
x
→
in
ℝ
n
such that
x
j
+
2
=
x
j
+
x
j
+
1
. for all
j
=
1
,
...
,
n
−
2
. Sec Exercise 73. Note that, by definition, any n consecutive Fibonacci numbers form a vector in V. Consider the basis
v
→
,
w
→
of V with
v
→
=
[
f
0
f
1
⋮
f
n
−
2
f
n
−
1
]
=
[
0
1
1
⋮
f
n
−
2
f
n
−
1
]
,
w
→
=
[
f
−
n
+
1
f
−
n
+
2
⋮
f
−
1
f
0
]
=
[
f
n
−
1
−
f
n
−
2
⋮
1
0
]
(In Exercise 4.3.7c, we introduce this basis in the case
n
=
4
.) We are told that
‖
v
→
‖
2
=
‖
w
→
‖
2
=
f
n
−
1
f
n
. (For extra credit, give a proof by induction on n.)
c. Find the basis
v
→
,
w
→
in the case
n
=
6
. Verify the identity
‖
v
→
‖
2
=
‖
w
→
‖
2
=
=
f
5
f
6
. Also, show that
v
→
is orthogonal to
w
→
.
d. Show that
v
→
is orthogonal to
w
→
for any even positive integer n.
e. For an even positive integer n, Iet P be the matrix of the orthogonal projection onto V. Show that the first column of P is
1
f
n
w
→
while the last column is
1
f
n
v
→
. Recall from Exercise 73 that P is a Hankel matrix, and note that a Hankel matrix is determined by its first and last columns Conclude that
P
=
1
f
n
[
f
−
n
+
1
f
−
n
+
2
⋯
f
−
1
f
0
f
−
n
+
2
f
−
n
+
3
⋯
f
0
f
1
⋮
⋮
⋱
⋮
⋮
f
−
1
f
0
⋯
f
n
+
3
f
n
−
2
f
0
f
1
⋯
f
n
−
2
f
n
−
1
]
meaning that the ijth entry of P is
f
i
+
j
−
n
−
1
f
n
.
f. Find the matrix P in the case
n
=
6
.