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Qualifying Examination
January, 1997
Math 554
In answering any part of a question you may assume the preceding parts.
Notation:
V
is a ﬁnite dimensional vector space over a ﬁeld
K
;
α
:
V
→
V
is a linear operator.
1.
In some basis of
V
,
α
is given by the matrix
A
=
11

20
21
02
10
0

121
. Find:
(1) the rational normal form of
α
.[
8
]
(2) the Jordan normal form of
α
7
]
2.
Let
P
be the space of polynomials of degree
<n
over
K
,andlet
δ
:
P
P
be
the operator, given by diﬀerentiation:
δ
(∑
n

1
i
=0
a
i
x
i
)
=
∑
n

1
i
=1
ia
i
x
i

1
.
Find the Jordan normal form of the
δ
2
,when
(1)
K
is the ﬁeld
C
of complex numbers.
[5]
(2)
K
is the ﬁeld
F
3
with 3 elements.
[5]
3.
A
α
invariant subspace
W
≤
V
is called irreducible, if the only proper
α

invariant subspaces of
W
are 0 and
W
itself.
(1) Prove that if the characteristic polynomial of
α
has an irreducible factor of
degree
d
,then
α
has an irreducible invariant subspace of dimension
d
.
[10]
(2) Prove the converse of (1).
[10]
4.
Let
v
1
,v
2
and
w
1
,w
2
be two pairs of vectors in a real inner product space
V
.
(1) Prove that if

v
1
=
w
1
,
v
2
=
w
2
,and
∠
(
v
1
2
)=
∠
(
w
1
2
), then there is
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 Vector Space

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