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§
2.4, Problem 7
Let
A
be an
n
×
n
matrix.
Theorem.
If
A
2
=0
,then
A
is not invertible
Proof.
For a contradiction assume that
A
2
=0and
A
is invertible. Then
A
has
an inverse
A

1
and
A
2
,
A

1
AA
=
A

1
0
,
IA
,
A
.
But then
A

1
A
=
A

1
0=0
6
=
I
– Contradiction.
Theorem.
If
AB
for some nonzero
n
×
n
matrix
B
A
is not invertible.
Proof.
Again, by contradiction. Assume
AB
A
is invertible. Then
AB
,
A

1
AB
=
A

1
0
,
IB
,
B
.
Contradiction: we assumed
B
was nonzero.
§
2.4, Problem 15
Theorem.
Let
V
and
W
be Fnitedimensional vector spaces, and let
T
:
V
→
W
be a linear transformation. Suppose that
β
=
{
β
1
,...,β
n
}
is a basis for
V
.Then
T
is an isomorphism if and only if
T
(
β
)
is a basis for
W
.
Proof.
(
⇒
) First assume that
T
is an isomorphism. Since
T
an isomorphism it
is onetoone and onto. Problem 14(c) on page 75 applies to tell us that
T
(
β
)
is a basis.
(
⇐
) Conversely, assume that
T
(
β
)isabas
isfor
W
. (We want to prove that
T
is an isomorphism. Since we already know
T
is linear we need prove that
T
is onetoone and onto.)
First we’ll prove
T
is onetoone. Let
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This note was uploaded on 11/30/2009 for the course MATH 115A taught by Professor Liu during the Winter '07 term at UCLA.
 Winter '07
 Liu
 Linear Algebra, Algebra

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