Math1030: HW1 solution
HW1: Sect 1.2 p15, Q12, Q16 Q19, Q20, Q21
12. Let
E
denote the set of even functions. For any
f,g
∈
E
,
f
(

t
) =
f
(
t
)
, g
(

t
) =
g
(
t
)
,
(
f
+
g
)(

t
) =
f
(

t
) +
g
(

t
)
=
f
(
t
) +
g
(
t
) = (
f
+
g
)(
t
)
,
∀
t
∈
R
.
So (
f
+
g
)
∈
E
. Addition is welldeﬁned in
E
.
For any
f
∈
E
,
c
∈
R
,
(
cf
)(

t
) =
c
[
f
(

t
)] =
c
[
f
(
t
)] = (
cf
)(
t
)
.
So (
cf
)
∈
E
. Scalar multiplication is welldeﬁned in
E
.
Let
f,g,h
∈
E
,
a,b,t
∈
R
,
(VS 1) (
f
+
g
)(
t
) =
f
(
t
) +
g
(
t
) =
g
(
t
) +
f
(
t
) = (
g
+
f
)(
t
).
(VS 2) [(
f
+
g
)+
h
](
t
) = (
f
+
g
)(
t
)+
h
(
t
) =
f
(
t
)+
g
(
t
)+
h
(
t
) =
f
(
t
) + (
g
+
h
)(
t
) = [
f
+ (
g
+
h
)](
t
).
(VS 3)
Denote f
0
(
t
) = 0
,
∀
t
∈
R
, then
(
f
+
f
0
)(
t
) =
f
(
t
)
,
∀
f
∈
E
.
(VS 4)
∀
f
∈
E
, denote
˜
f
(
t
) =

f
(
t
), then
˜
f
(

t
) =

f
(

t
) =

f
(
t
) =
˜
f
(
t
)
,
∀
˜
f
∈
E
, and
˜
f
+
f
= 0.
(VS 5) (1
f
)(
t
) = 1
f
(
t
) =
f
(
t
).
(VS 6) [(
ab
)
f
](
t
) =
abf
(
t
) =
a
(
bf
)(
t
) = [
a
(
bf
)](
t
).
(VS 7) [
a
(
f
+
g
)](
t
) =
a
(
f
+
g
)(
t
) =
a
[
f
(
t
) +
g
(
t
)] =
af
(
t
) +
ag
(
t
) = (
af
+
ag
)(
t
).
(VS 8) [(
a
+
b
)
f
](
t
) = (
a
+
b
)
f
(
t
) =
af
(
t
) +
bf
(
t
) = (
af
+
bf
)(
t
).
So
E
is a vector space.
16.
V
is a vector space over
F
with the usual deﬁnitions of matrix
addition and scalar multiplication. Let
A,B
∈
V
,
c
∈
F
, then
A
+
B
∈
V
,
cA
∈
V
. So addition and scalar multiplication is
welldeﬁned in
V
. Check (
V S
1)

(
V S
8) hold in the same way
as question 12.