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Qualifying Examination
August, 1996
Math 554
In answering any part of a question you may assume the preceding parts.
Notation:
K
is a ﬁeld;
M
n
(
K
) is the set of
n
×
n
matrices with elements from
K
;
V
is an
n
dimensional vector space over
K
;
α
is a linear operator on
V
.
1.
Prove that if
A,B
∈
M
n
(
K
) and one of
is invertible, then det(
aA
+
B
)=0
for at most
n
distinct values of
a
∈
K
.[
8
p
o
i
n
t
s
]
2.
Let
A
T
denote the transpose of
A
∈
M
n
(
K
). Prove that there exists an invertible
P
∈
M
n
(
K
), such that
PAP

1
=
A
T
8
p
o
i
n
t
s
]
3.
Let
π
1
and
π
2
be linear operators on a vector space
V
, such that
π
1
π
2
=
π
2
π
1
π
2
1
=
π
1
π
2
2
=
π
2
.
Prove that
V
is the direct sum of the following four subspaces:
[8 points]
Im
π
1
∩
Im
π
2
Im
π
1
∩
Ker
π
2
Ker
π
1
∩
Im
π
2
Ker
π
1
∩
Ker
π
2
.
4.
Prove that if
α
has the same matrix in all bases of
V
, then there exists an
a
∈
K
such that
α
=
a
id
V
8
p
o
i
n
t
s
]
5.
Prove that if rank(
α
) = 1, then the minimal polynomial of
α
has the form
x
(
x

a
)forsome
a
∈
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