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80
Chapter 5
17. (a)
A
is symmetric since
A
T
=(
xy
T
+
yx
T
)
T
xy
T
)
T
+(
T
)
T
y
T
)
T
x
T
x
T
)
T
y
T
=
T
+
xy
T
=
A
(b)
For any vector
z
in
R
n
A
z
=
xy
T
z
+
yx
T
z
=
c
1
x
+
c
2
y
where
c
1
=
y
T
z
and
c
2
=
x
T
z
.If
z
is in
N
(
A
) then
0
=
A
z
=
c
1
x
+
c
2
y
and since
x
and
y
are linearly independent we have
y
T
z
=
c
1
= 0 and
x
T
z
=
c
2
= 0. So
z
is orthogonal to both
x
and
y
. Since
x
and
y
span
S
it follows that
z
∈
S
⊥
.
Conversely, if
z
is in
S
⊥
then
z
is orthogonal to both
x
and
y
.It
follows that
A
z
=
c
1
x
+
c
2
y
=
0
since
c
1
=
y
T
z
= 0 and
c
2
=
x
T
z
= 0. Therefore
z
is in
N
(
A
) and hence
N
(
A
)=
S
⊥
.
(c)
Clearly dim
S
= 2 and by Theorem 5.2.2, dim
S
+ dim
S
⊥
=
n
. Using
our result from part (a) we have
dim
N
(
A
) = dim
S
⊥
=
n

2
So
A
has nullity
n

2. It follows from the RankNullity Theorem that
the rank of
A
must be 2.
SECTION 3
1.
(b)
A
T
A
=
6

1

16
and
A
T
b
=
20

25
The solution to the normal equations
A
T
A
x
=
A
T
b
is
x
=
19
/
7

26
/
7
2.
(Exercise 1b.)
(a)
p
=
1
7
(

45
,
12
,
71)
T
(b)
r
=
1
7
(115
,
23
,
69)
T
(c)
A
T
r
=

12 1
11

2
115
7
23
7
69
7
=
0
0
0
Therefore
r
is in
N
(
A
T
).
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81
6.
A
=
1

11
10
0
1
12
4
,
b
=
0
1
3
9
A
T
A
=
42
6
26
8
681
8
,A
T
b
=
13
21
39
The solution to
A
T
A
x
=
A
T
b
is (0
.
6
,
1
.
7
,
1
.
2)
T
. Therefore the best
least squares Ft by a quadratic polynomial is given by
p
(
x
)=0
.
6+1
.
7
x
+1
.
2
x
2
7.
To Fnd the best Ft by a linear function we must Fnd the least squares
solution to the linear system
1
x
1
1
x
2
.
.
.
.
.
.
1
x
n
c
0
c
1
=
y
1
y
2
.
.
.
y
n
If we form the normal equations the augmented matrix for the system will
be
n
n
±
i
=1
x
i
n
±
i
=1
y
i
n
±
i
=1
x
i
n
±
i
=1
x
2
i
n
±
i
=1
x
i
y
i
If
x
= 0 then
n
±
i
=1
x
i
=
n
x
=0
and hence the coeﬃcient matrix for the system is diagonal. The solution is
easily obtained.
c
0
=
n
±
i
=1
y
i
n
=
y
and
c
1
=
n
±
i
=1
x
i
y
i
n
±
i
=1
x
2
i
=
x
T
y
x
T
x
82
Chapter 5
8.
To show that the least squares line passes through the center of mass, we
introduce a new variable
z
=
x

x
. If we set
z
i
=
x
i

x
for
i
=1
,...,n
,
then
z
= 0. Using the result from Exercise 7 the equation of the best least
squares Ft by a linear function in the new
zy
coordinate system is
y
=
y
+
z
T
y
z
T
z
z
If we translate this back to
xy
coordinates we end up with the equation
y

y
=
c
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This note was uploaded on 04/18/2010 for the course FINANCE 1231854365 taught by Professor Wuyiling during the Spring '10 term at Nashville State Community College.
 Spring '10
 wuyiling

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