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Math 121 Homework, Week 1
Michael Von Korﬀ
October 1, 2004
Problem 1.
A realvalued function
f
deﬁned on the real line is called an even
function if
f
(

x
) =
f
(
x
)
for each real number
x
. Prove that the set of even
functions deﬁned on the real line with the operations of addition and scalar
multiplication deﬁned in Example 3 is a vector space.
Proof.
We could show each of the 8 vector space requirements as well as closure
under addition and scalar multiplication; together, all of these would show that
the set is a vector space. But there’s an easier way:
Let S be the set of even, realvalued functions. Then S is a subset of the
set of all realvalued functions, which is a vector space by Example 3. Thus, in
order to show that S is a vector space, we need only show that it is a subspace
of the set of all realvalued functions. To do this, we need only apply Theorem
1.3 and show that S is closed under addition and scalar multiplication. That is,
we must show that
∀
f,g
∈
S,
∀
λ
∈
R
,f
+
g
∈
S
and
λf
∈
S
.
But
∀
x
∈
R
,
(
f
+
g
)(
x
) =
f
(
x
) +
g
(
x
)
=
f
(

x
) +
g
(

x
)
= (
f
+
g
)(

x
)
so
f
+
g
is even, and S is closed under addition. Similarly,
(
λf
)(
x
) =
λ
(
f
(
x
)) =
λ
(
f
(

x
))
= (
λf
)(

x
)
so
λf
is even, and S is closed under scalar multiplication. Thus S is a vector
subspace of the set of realvalued functions.
Problem 2.
Let V denote the set of all
m
×
n
matrices with real number entries.
V is a vector space over the ﬁeld
R
of real numbers under the usual deﬁnitions
of matrix addition and multiplication. Is V a vector space over the ﬁeld
Q
of
rational numbers under the same addition and multiplication?
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This note was uploaded on 02/22/2012 for the course EE 441 taught by Professor Neely during the Spring '08 term at USC.
 Spring '08
 Neely

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