QUALIFYING
EXAMINATION
JANUARY
2002
MATH
554

PROF.
MATSUKI
Name
ID
#
1. Let
A
be an Abelian group satisfying conditions
(
α
) (5
3
·
7
4
)
A
= 0, and
(
β
) #
{
a
∈
A
; (5
·
7)
a
= 0
}
= 5
2
·
7
2
.
(i) Show that
A
is finitely generated.
(10 points)
(ii) How many such Abelian groups that satisfy conditions (
α
) and (
β
) do we
have up to isomorphism ?
(10 points)
2. Let
A
be the following 3
×
3 matrix
A
=
0
2
1

4
6
2
4

4
0
.
(i) Find the rational canonical form
C
of
A
, and an invertible matrix
S
such
that
C
=
S

1
AS
.
(10 points)
(ii) Find the Jordan canonical form
J
of
A
, and an invertible matrix
T
such
that
J
=
T

1
AT
.
(10 points)
3. Let
A
be a 3
×
3 matrix whose Jordan canonical form is

2
0
0
0
7
1
0
0
7
.
Let
B
be a 4
×
4 matrix whose Jordan canonical form is
7
1
0
0
0
7
1
0
0
0
7
0
0
0
0
7
.
Consider the space
V
of 3
×
4 matrices
X
with entries in
C
such that
AX
=
XB.
Find the dimension of the vector space
V
over
C
.
(20 points)
Typeset by
A
M
S
T
E
X
1
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2
4. Let
V
=
C
4
be a 4dimensional vector space over
C
and let
f
:
V
→
V
be a
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 Spring '09
 Math, Linear Algebra, Matrices, Vector Space, Jordan Canonical Form

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