Solutions to PS 5 (Math 121)
J. Scholtz
November 15, 2006
Question 1: 2.5/7
in
R
2
let
L
be a line
y
=
mx
,
m
6
= 0. Find an expression for the following
T
(
x,y
)’s. First let us ﬁgure
out a change of basis matrix from the standard basis to the basis
a
=
{
(
m,
1)
,
(

1
,m
)
}
, because that
is the basis with a vector along the line and a vector perpendicular to the line. We need to express the
standard basis vectors in terms of the new basis vectors: (1
,
0) =
1
m
2
+1
(
m
(
m,
1)

(

1
,m
)) and similarly
(0
,
1) =
1
m
2
+1
((
m,
1) +
m
(

1
,m
)) Then the change of matrix
P
will look like:
P
=
1
m
2
+ 1
±
m
1

1
m
²
Then the inverse to
P
is:
P

1
=
1
m
2
+ 1
±
m

1
1
m
²
part a) A reﬂection along
L
The reﬂection along the new basis just takes a form:
[
T
]
β
0
=
±
1
0
0

1
²
So in the standard basis:
[
T
]
β
=
P

1
AP
=
1
m
2
+ 1
±
m
2
+
m m
+ 1
m

m
2
1

m
²
part b) A projection onto
L
Similarly in the
β
0
basis:
[
T
]
β
0
=
±
1
0
0
0
²
Hence in the
β
standard basis:
[
T
]
β
=
P

1
AP
=
1
m
2
+ 1
±
m
2
m
m
1
²
Question 2: 2.6/13
Let us ﬁrst deﬁne an
annihilator
S
0
of a subset of a vector space
S
⊂
V
.
S
0
=
{
f
∈
V
*
:
f
(
x
) = 0
∀
x
∈
S
}
1.
Prove that
S
0
is a subspace of
V
*
.
1
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View Full Document(a) Clearly 0
∈
S
0
.
(b) If
a
(
x
)
,b
(
x
)
∈
S
0
, then this means that
a
(
x
) =
b
(
x
) = 0 for all
x
∈
S
. Therefore so will
(
a
+
b
)(
x
).
(c) If
a
(
x
)
inS
0
so will
ca
(
x
) by the same argument.
2.
If
W
⊂
V
and
x /
∈
W
, then show there exists
f
∈
W
0
such that
f
(
x
)
6
= 0
.
Pick a basis for
W
,
β
W
=
{
v
1
,...,v
k
}
and extend it onto
V
,
β
V
=
{
v
1
,...,v
k
,v
k
+1
,...,v
n
}
. Then
form a dual basis to this basis: (
β
V
)
*
=
{
f
1
,...,f
k
,f
k
+1
,...,f
n
}
. Clearly
W
0
∩{
f
1
,...,f
k
}
since
f
i
(
v
i
) = 1
6
= 0. On the other hand we know that
f
k
+1
,...,f
n
∈
W
0
, since
v
k
+1
,...,v
n
/
∈
W
. But
since
v /
∈
W
, then this means that
v
=
w
+
∑
a
i
v
i
, where
i > k
and at least one of the
a
i
’s is non
zero, let it be
a
l
. But then this means that
f
l
(
v
) =
a
l
6
= 0, therefore we have found one.
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 Spring '08
 GUREVITCH
 Math

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