Math 115A
Homework 2
Due October 24th, 2008
1. For each of the following maps
T
:
V
→
W
, check if
T
is an
F
linear transformation. If
yes, write out a proof. If not, prove that.
(2 pts each)
a)
F
=
R
,
V
=
R
3
,
W
=
R
2
and
T
(
a
1
,a
2
,a
3
) = (
a
1
a
2
,a
3
+ 2
a
1
).
Let
λ
= 2
∈
R
and
a
= (
a
1
,a
2
,a
3
) = (1
,
1
,
1). Then
T
(
λ
a
) =
T
(2
,
2
,
2) = (4
,
6) but
λT
(
a
) = (2
,
6). That is,
T
does not commute with scalar multiplication and is therefore not
a linear transformation.
b)
F
=
C
,
V
=
C
2
,
W
=
C
and
T
(
z,w
) =
z
.
Let
λ
=
i
∈
C
and
v
= (1
,
0)
∈
V
. Then
T
(
λ
v
) =

i
but
λT
(
v
) =
i
. That is,
T
does not
commute with scalar multiplication and is therefore not a linear transformation.
c)
F
=
R
,
V
=
C
(
R
) the set of continuous realvalued functions on the real numbers,
W
=
R
and
T
(
f
) =
R
1
0
x
2
f
(
x
)
dx
.
The map
U
:
V
→
V
that is deﬁned by
U
(
f
)(
x
) =
x
2
f
(
x
) is a linear transformation
because multiplication of real numbers is commutative, and multiplication and addition are
distributive. Moreover, the map
S
:
V
→
W
=
R
deﬁned by
S
(
f
) =
R
1
0
f
(
x
)
dx