MATH 110: LINEAR ALGEBRA
SPRING 2007/08
PROBLEM SET 3 SOLUTIONS
1.
Let
W
1
,...,W
n
be nontrivial subspaces of a vector space
V
over
R
. Show that there exists a
vector
v
∈
V
such that
v
/
∈
W
i
for all
i
= 1
,...,n
.
Solution.
If
W
1
⊆
W
2
∪ ··· ∪
W
n
, then any
v
/
∈
W
i
for all
i
= 2
,...,n
will automatically
satisfy
v
/
∈
W
1
. If
W
2
∪ ··· ∪
W
n
⊆
W
1
, then any
v
/
∈
W
1
will automatically satisfy
v
/
∈
W
i
for
all
i
= 2
,...,n
. Hence we may assume without loss of generality that
W
1
*
W
2
∪ ··· ∪
W
n
and
W
2
∪ ··· ∪
W
n
*
W
1
.
Let
v
1
∈
W
1
and
v
1
/
∈
W
2
∪ ··· ∪
W
n
. Let
v
2
∈
W
2
∪ ··· ∪
W
n
and
v
2
/
∈
W
1
. Consider the set
L
=
{
λ
v
1
+
v
2

λ
∈
R
}
.
For any
λ
,
λ
v
1
+
v
2
/
∈
W
1
since otherwise
v
2
= (
λ
v
1
+
v
2
)

λ
v
1
∈
W
1
— a contradiction. Now
let
i
∈ {
2
,...,n
}
. If
λ
i
v
1
+
v
2
∈
W
i
for some
λ
i
, then for any
λ
6
=
λ
i
,
λ
v
1
+
v
2
/
∈
W
i
since
otherwise
v
1
= (
λ

λ
i
)

1
[(
λ
v
1
+
v
2
)

(
λ
i
v
1
+
v
2
)]
∈
W
i
— a contradiction. Hence each
W
i
contains at most one element of
L
. Since
L
has inﬁnitely many elements, there exists
v
∈
L
such that
v
/
∈
W
i
for all
i
= 1
,...,n
.
2.
Let
W
1
,...,W
n
be subspaces of a vector space
V
. Recall from Problem Set
2
that
W
1
+
···
+
W
n
is a
direct sum
(and hence may be denoted
W
1
⊕ ··· ⊕
W
n
) if
W
i
∩
±
∑
j
6
=
i
W
j
²
=
{
0
}
for all
i
= 1
,...,n.
Show that the following statements are equivalent.
(i)
W
1
+
···
+
W
n
is a direct sum.
(ii) The function deﬁned by
f
:
W
1
× ··· ×
W
n
→
V,
f
(
w
1
,...,
w
n
) =
w
1
+
···
+
w
n
is injective.
(iii)
W
1
,...,W
n
satisﬁes
(
W
1
+
···
+
W
i
)
∩
W
i
+1
=
{
0
}
for all
i
= 1
,...,n

1
.
Solution.
We will make use of the three equivalent statements characterizing direct sum that
we proved in Problem Set
2
.
(i)