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MATH 225
SAMPLE FINAL EXAM PROBLEMS
1. Can the matrix
A
exist or not exist:
1
2
1
1
1
2
1
1
1
1
1
1
1
1
1
0
0
0
1
1
0
0
0
1
2
"
#
$
$
$
$
$
$
%
'
'
'
'
'
'
A
1
0
0
1
6
7
1
7
8
"
#
$
$
$
%
'
'
'
=
1
0
1
2
1
1
3
0
1
4
1
1
5
1
2
"
#
$
$
$
$
$
$
%
'
'
'
'
'
'
?
2. Let
U
and
V
be subspaces of
R
10
x
1
so that {
α
1
,
2
,
3
,
4
,
5
,
6
} is a basis of
U
and {
β
1
,
2
,
3
,
4
,
5
}
is a basis of
V
.
Argue that there is a nonzero
γ
∈
U
∩
V
.
3. Let
T
:
V
→
V
be a linear transformation of the vector space
V
over
R
and set
W
= {
∈
V
| for
k
≥
1
and depending
on
,
T
k
(
) = 0
V
}.
Show that
W
is a subspace of
V
.
If
T
m
(
) = 0 but
T
m
–1
(
)
≠
0 show that {
,
T
(
),
T
2
(
), . . . ,
T
m
–1
(
)} is linearly independent.
4. Determine if
A
=
3
0
!
2
!
2
1
2
1
0
0
"
#
$
$
%
'
'
∈
M
3
(
R
) is diagonalizable.
If it is not, give reasons why.
If
A
is
diagonalizable find
P
∈
M
3
(
R
) and a diagonal
D
∈
M
3
(
R
) so that
P
–1
AP
=
D
.
5. Let

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