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2.3.
if
i
6
=
j
, then
A
ij
= 0, thus
A
ij
=
δ
ij
A
ij
and if
i
=
j
, then
δ
ij
= 1, therefore
A
ij
=
δ
ij
A
ij
11.
(
⊆
)
∀
v
∈
V
,
T
(
V
)
∈
R
(
T
), since
T
(
T
(
V
)) =
{
0
}
,
T
(
V
)
∈
N
(
T
)
∴
R
(
T
)
⊆
N
(
T
)
(
⊇
)
∀
v
∈
V
,
T
(
V
)
⊆
R
(
T
),
∴
R
(
T
)
⊆
N
(
T
)
∴
T
2
(
v
)
⊆
T
(
T
(
v
)
⊆
T
(
N
(
T
)) = 0
∴
T
2
=
T
0
12.
(a) Assume
UT
:
V
→
Z
is onetoone and let
T
(
V
1
) =
T
(
v
2
) for
v
1
,v
2
∈
V
Then
U
(
T
(
v
1
)) =
U
(
T
(
v
2
)) i.e (
UT
)(
v
1
) = (
UT
)(
v
2
)
Since
UT
is onetoone,
v
1
=
v
2
∴
T is onetoone
(Example)
V
=
R
2
, W
=
Z
=
R
3
T
:
V
→
W T
(
a,b
) = (
a,b,
0)
U
:
W
→
Z U
(
a,b,c
) = (
a,b,
0)
(b) Assume that
UT
:
V
→
Z
is onto and let
z
∈
Z
94
PNUMATH
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View Full Document 2.3.
∃
v
∈
V
s.t (
UT
)(
v
) =
z
Let
w
=
T
(
v
)
Then
w
∈
W
and
U
(
W
) =
U
(
T
(
v
)) = (
UT
)(
v
) =
z
∴
U
:
W
→
Z
is onto
(Example)
V
=
W
=
R
3
, W
=
R
2
T
:
V
→
W T
(
a,b,c
) = (
a,b,
0)
U
:
W
→
Z U
(
a,b,c
) = (
a,b
)
UT
is onetoone but
U
is not
(c) (1) Let
v
1
,v
2
∈
V
and assume (
UT
)(
v
1
) = (
UT
)(
v
2
)
If
U
(
T
(
v
1
)) =
U
(
T
(
v
2
)), then
T
(
v
1
) =
T
(
v
)
2
(
∵
U is onetoone)
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This note was uploaded on 02/14/2012 for the course MAS 4105 taught by Professor Rudyak during the Spring '09 term at University of Florida.
 Spring '09
 RUDYAK

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