2.1.
§
2. Linear Transformations and Matrices
2.1. Linear Transformations, Null Spaces, and Ranges
1. (a) T
(b) F (
∵
) If
∀
x,y
∈
V
and
c
∈
F
, T(
x
+
y
) =T(
x
)+T(
y
) and T(
cx
) =
c
T(
x
),
then T is a linear transformation
(c) F (
∵
) T is linear and onetoone if and only if
(d) T (
∵
) T(0
V
) =T(0
V
+ 0
V
) =T(0
V
)+T(0
V
)
∴
T(0
V
) = 0
W
(e) F (
∵
) p.70 Theorem 2.3
(f) F (
∵
) T is linear and one to one, then
(g) T (
∵
) p.73 Corollary to Theorem 2.6
(h) F (
∵
) p.72 Theorem 2.6
2.
(1)
T
((
a
1
,a
2
,a
3
) + (
b
1
,b
2
,b
3
)) =
T
(
a
1
+
b
1
,a
2
+
b
2
,a
3
+
b
3
)
= ((
a
1
+
b
1
)

(
a
2
+
b
2
)
,
2(
a
3
+
b
3
))
= (
a
1

a
2
)
,
2
a
3
) + (
b
1

b
2
)
,
2
b
3
)
=
T
(
a
1
,a
2
,a
3
) +
T
(
b
1
,b
2
,b
3
)
T
(
c
(
a
1
,a
2
,a
3
)) =
T
(
ca
1
,ca
2
,ca
3
) = (
ca
1

ca
2
,
2
ca
3
) =
c
(
a
1

a
2
,
2
a
3
) =
cT
(
a
1
,a
2
,a
3
)
Thus, T is linear.
(2) N(T)=span
{
(1
,
1
,
0)
}
R(T)=span
{
(1
,
0)
,
(0
,
1)
}
55
PNUMATH