Math 235
Assignment 4 Solutions
1.
Prove that the product of two orthogonal matrices is an orthogonal matrix.
Solution: Let
P
and
Q
be orthogonal matrices. Then we have
(
PQ
)
T
(
PQ
) =
Q
T
P
T
PQ
=
Q
T
Q
=
I,
since
P
T
P
=
I
and
Q
T
Q
=
I
. Thus
PQ
is also orthogonal.
2.
Observe that the dot product of two vectors
~x, ~
y
∈
R
n
can be written as
~x
·
~
y
=
~x
T
~
y.
Use this fact to prove that if an
n
×
n
matrix
R
is orthogonal, then
k
R~x
k
=
k
~x
k
for every
~x
∈
R
n
.
Solution: Suppose that
R
is orthogonal, then
R
T
R
=
I
. Then for any
~x
∈
R
n
we have
k
R~x
k
2
= (
R~x
)
·
(
R~x
) = (
R~x
)
T
(
R~x
) = (
~x
T
R
T
)(
R~x
) =
~x
T
(
R
T
R
)
~x
=
~x
T
~x
=
k
~x
k
2
.
Hence
k
R~x
k
=
k
~x
k
for every
~x
.
3.
Suppose that
B
=
{
~v
1
,~v
2
,~v
3
}
is a basis for a real inner product space
V
. Define a 3
×
3
matrix
G
by
(
G
)
ij
=
h
~v
i
,~v
j
i
a) Prove that
G
is symmetric (
G
T
=
G
).
Solution: Because the inner product is symmeric,
(
G
)
ji
=
h
~v
j
,~v
i
i
=
h
~v
i
,~v
j
i
= (
G
)
ij
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2
b) Show that if [
~x
]
B
=
x
1
x
2
x
3
and [
~
y
]
B
=
y
1
y
2
y
3
, then
h
~x, ~
y
i
=
x
1
x
2
x
3
G
y
1
y
2
y
3
Solution: By the bilinearity of
h
,
i
h
~x, ~
y
i
=
h
x
1
~v
1
+
x
2
~v
2
+
x
3
~v
3
, y
1
~v
1
+
y
2
~v
2
+
y
3
~v
3
i
=
x
1
y
1
h
~v
1
,~v
1
i
+
x
1
y
2
h
~v
1
,~v
2
i
+
· · ·
+
x
3
y
3
~v
3
i
=
(
x
1
(
G
)
11
+
x
2
(
G
)
21
+
x
3
(
G
)
31
)
y
1
+
(
x
1
(
G
)
12
+
x
2
(
G
)
22
+
x
3
(
G
)
32
)
y
2
+
+
(
x
1
(
G
)
13
+
x
2
(
G
)
23
+
x
3
(
G
)
33
)
y
3
=
x
1
x
2
x
3
G
y
1
y
2
y
3
c) Consider the inner product
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 Spring '10
 WILKIE
 Matrices, Dot Product, Orthogonal matrix, inner product, Inner product space, real inner product

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