Math 115a
Midterm 1
Lecture 1
Fall 2009
Name:
Signature:
Instructions:
•
There are 5 problems. Make sure you are not missing any pages.
•
Unless stated otherwise, you may use without proof anything proven in the
sections of the book covered by this test.
•
You may only cite an exercise from the book if it was assigned as homework.
Question
Points
Score
1
10
2
10
3
10
4
15
5
15
Total:
60
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1.
(10 points) Recall that
P
5
(
R
)
is the vector space of polynomials of degree less
than or equal to 5. Let
A
=
{
f
∈
P
5
(
R
):
f
(0) = 0
}
and
B
=
{
f
∈
P
5
(
R
):
f
(1) = 0
}
.
Prove that
A
∩
B
is a subspace of
P
5
(
R
)
.
Solution:
Let
p
(
x
) = 0 for all
x
∈
R
. Certainly
p
(0) = 0, so
p
∈
A
. Also,
p
(1) = 0, so
p
∈
B
.
Hence
p
∈
A
∩
B
. This proves that the zero polynomial is in
A
∩
B
. Now let
f, g
∈
A
∩
B
and
λ
∈
R
. We know
(
f
+
g
)(
x
) =
f
(
x
) +
g
(
x
)
for any
x
∈
R
, by the definition of addition in
P
5
(
R
). In particular,
(
f
+
g
)(0) =
f
(0) +
g
(0) = 0 + 0 = 0
so
f
+
g
∈
A
, and
(
f
+
g
)(1) =
f
(1) +
g
(1) = 0 + 0 = 0
so
f
+
g
∈
B
. Hence
f
+
g
∈
A
∩
B
. Similarly,
(
λf
)(
x
) =
λf
(
x
)
for any
x
∈
R
, by the definition of scalar multiplication in
P
5
(
R
). In particular,
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 Fall '06
 Taft
 Math, Linear Algebra, Algebra, Vector Space, linearly independent subset

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