1
Definition.
Given a vector space
V
, a function
T
:
V
→
V
is a
linear
operator
if for all
u
,
v
∈
V
and any scalar
c
,
T
(
u
+
v
) =
T
(
u
) +
T
(
v
) and
T
(
c
u
) =
cT
(
u
).
Example:
U
:
R
m
×
n
→
R
n
×
m
is defined by
U
(
A
) =
A
T
⇒
U
is a linear transformation, since
(
A
+
B
)
T
=
A
T
+
B
T
and (
cA
)
T
=
cA
T
,
∀
A
,
B
∈
R
m
×
n
.
Example:
C
([
a
,
b
]) = {
f

f
: [
a
,
b
]
→
R
,
f
is continuous}
⊆
F
([
a
,
b
]).
⇒
C
([
a
,
b
]) is a subspace.
T
:
C
([
a
,
b
])
→
R
is defined by
T
(
f
) =
⇒
T
is a linear operator on
C
([
a
,
b
]) , since
∀
f
,
g
∈
C
([
a
,
b
]),
T
(
f
+
g
) =
T
(
f
) +
T
(
g
), and
∀
c
∈
R
,
T
(
cf
) =
cT
(
f
).
Example:
C
∞
= {
f

f
:
R
→
R
,
f
has derivatives of all order}
⊆
F
(
R
).
⇒
C
∞
is a subspace.
D
:
C
∞
→
C
∞
is defined by
D
(
f
) =
f
′
for all
f
∈
C
∞
.
⇒
D
is a linear operator on
C
∞
, since for all
f
,
g
∈
C
∞
,
D
(
f
+
g
) = (
f
+
g
)
′
=
f
′
+
g
′
=
D
(
f
) +
D
(
g
),
and for all
c
∈
R
,
D
(
cf
) =
cf
′
=
cD
(
f
).
Example: Let