1
Math 110 Homework 3
Partial Solutions
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1.6.23 In this case, by a previous homework problem, we already have that
W
1
⊆
W
2
.
(a) I claim that dim(
W
1
)=dim(
W
2
) if and only if
W
1
=
W
2
if and only
if
v
∈
span(
{
v
1
,...,v
k
}
). The ﬁrst if and only if is straightforward.
For the second, suppose ﬁrst that
W
1
=
W
2
. Then
v
∈
W
1
. This
proves that
v
is in the span of
{
v
1
,...,v
k
}
. On the other hand,
suppose that
v
∈
span(
{
v
1
,...,v
k
}
). We will then show that
W
2
⊆
W
1
and thus that they are equal. Let
x
∈
W
2
. Then there exist
scalars
a
1
,...,a
k
,a
k
+1
∈
F
such that
x
=
a
1
v
1
+
···
+
a
k
v
k
+
a
k
+1
v
.
But
v
∈
span(
{
v
1
,...,v
k
}
), so there are
b
1
,...,b
k
∈
F
such that
v
=
b
1
v
1
+
···
+
b
k
v
k
. Then
x
=
a
1
v
1
+
···
+
a
k
v
k
+
a
k
+1
(
b
1
v
1
+
···
+
b
k
v
k
is in span(
{
v
1
,...,v
k
}
).