Problem
1
2
3
4
5
Bonus:
Total:
Points
6
12
10
10
12
10
50+10
Scores
Mat 310 – Linear Algebra – Fall 2004
Name:
Id. #:
Lecture #:
Test 2
(November 05 / 60 minutes)
There are 5 problems worth 50 points total and a bonus problem worth up to 10 points.
Show all work. Always indicate carefully what you are doing in each step; otherwise it may not be possible
to give you appropriate partial credit.
1.
[6 points] Let
W
1
and
W
2
be linear subspaces of a vector space
V
such that
W
1
+
W
2
=
V
and
W
1
∩
W
2
=
{
0
}
. Prove that for each vector
α
∈
V
there are
unique
vectors
α
1
∈
W
1
and
α
2
∈
W
2
such that
α
=
α
1
+
α
2
.
Solution:
Since
W
1
+
W
2
=
V
, every vector
α
∈
V
can be represented as
α
=
α
1
+
α
2
for some
α
1
∈
W
1
and
α
2
∈
W
2
.
We must then show that this representation is unique. Suppose that there are two other vectors
β
1
∈
W
1
and
β
2
∈
W
2
such that
α
=
β
1
+
β
2
.
Then
α

α
= 0 = (
α
1
+
α
2
)

(
β
1
+
β
2
)
so
α
1

β
1
=
β
2

α
2
.
Call the above vector
γ
. Then since
α
1
and
β
1
are both in
W
1
, their diﬀerence
γ
must also be in
W
1
.
Similarly,
γ
must be in
W
2
, and so
γ
∈
W
1
∩
W
2
.
Therefore
γ
= 0
, that is,
α
1
=
β
1
and
α
2
=
β
2
.
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[12 points] Consider the vectors in
R
4
deﬁned by
α
1
= (

1
,
0
,
1
,
2)
, α
2
= (3
,
4
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