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Math 235: Linear Algebra HW 8
Problem 3.6.1
Problem 3.6.1
Let
n
be a positive integer and
F
a ﬁeld. Let
W
be the set of all vectors (
x
1
,...,x
n
) in
F
n
such that
x
1
+
x
2
+
···
+
x
n
= 0.
(a)
Prove that
W
0
consists of all linear functaionals
f
of the form
f
(
x
1
,...,x
n
) =
c
n
X
j
=1
x
i
Let
S
=
(
f
(
x
1
,...,x
n
) =
c
n
X
i
=1
x
i
)
. Clearly for every
w
∈
W
and
f
∈
S
we have that
f
(
w
) =
c
n
X
i
=1
x
i
=
c
·
0 = 0
so
S
⊆
W
0
.
Now we must show that
S
is all of
W
0
, we’ll show this using dimensions.
dim (
W
) =
n

1 (we can pick
x
1
,...,x
n

1
then
x
n
=

n

1
X
i
=1
x
i
). We know that dim
W
+ dim
W
0
=
dim
F
n
=
n
⇒
n

1 + dim
W
0
=
n
⇒
dim
W
0
= 1. Clearly dim
S
= 1 so we get that
S
=
W
0
.
(a)
(b)
Show that the dual space
W
*
of
W
can be ’naturally’ identiﬁed with the linear functionals
f
(
x
1
,...,x
n
) =
c
1
x
1
+
···
+
c
n
x
n
on
F
n
which satisfy
c
1
+
···
+
c
n
= 0.
This is actually quite easy.
Let
w
= (
w
1
,...,w
n
)
∈
W
then
w
1
+
w
2
+
···
+
w
n
= 0, so we get the linear functional
f
=
w
1
x
1
+
w
2
x
2
+
···
+
w
n
x
n
.
Similarly taking a linear funcional
f
=
c
1
x
1
+
c
2
x
2
+
···
+
c
n
x
n
where
c
1
+
···
+
c
n
= 0 we get the
vector
w
= (
c
1
,...,c
n
) which is in
W
since
c
1
+
···
+
c
n
= 0.
(b)
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View Full Document Math 235: Linear Algebra HW 8
Problem 3.6.2
Problem 3.6.2
Use Theorem 20 to prove the following. If
W
is a subspace of a ﬁnitedimensional vector space
V
and if
{
g
1
,...,g
r
}
is any basis of
W
0
, then
W
=
r
\
i
=1
N
g
i
For every
f
∈
W
0
we have that
f
=
r
X
n
=1
a
i
g
i
so by theorem 20
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This note was uploaded on 11/14/2010 for the course PHYCS 498 taught by Professor Aa during the Spring '10 term at University of Illinois, Urbana Champaign.
 Spring '10
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