1
Lecture 5
5.1 Linear Transformation (Conti.)
Thm: Preservation of subspaces
Let
V
V
,
be vector spaces and
V
V
T
:
be a linear transformation.
1.
If W is a subspace of V, then T[W] is a subspace of
V
.
2.
If W
´
is a subspace of
V
, then T
1
[W] is a subspace of V.
5.2 Kernel and image of a linear transformation
Let
V
V
,
be vector spaces and
V
V
T
:
be a linear transformation.
ker(T) =
V
v
v
T
v
,
0
)
(
Range(T) =
V
v
v
T
)
(
Example: If
V
V
T
:
is a linear transformation, show that Ker(T) is a subspace of V
and range(T) is a subspace of
V
.
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Example: Define a transformation T: M
nn
→ M
nn
by T(A) = A
–
A
T
for all A in M
nn
.
Show that T is linear and that
a)
ker(T) consists of all symmetric matrices;
b)
range(T) consists of all skewsymmetric matrices.
Let
V
V
,
be vector spaces and
V
V
T
:
be a linear transformation.
Homogeneous equation
0
)
(
x
T
solution set = ker(T)
Nonhomogeneous equation
b
x
T
)
(
Solution set = {
h
h
p
ker(T)} if
b
p
T
)
(
for some
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 Fall '09
 YANG
 Linear Algebra, Algebra, Vector Space, Linear map, Isomorphism, linear transformation, MNN

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