QUALIFYING EXAMINATION
August 1998
MATH 554  Profs. Heinzer/Matsuki
Name:
(12) 1. Give an example of an inﬁnite dimensional vector space
V
over the ﬁeld of real
numbers
R
and linear operators
S
and
T
on
V
such that
(i)
S
is onto, but not onetoone.
(ii)
T
is onetoone, but not onto.
(10) 2. Let
Q
denote the ﬁeld of rational numbers. Give an example of a linear operator
T
:
Q
3
→
Q
3
having the property that the only
T
invariant subspaces are the
whole space and the zero subspace. Explain why your example has this property.
(18) 3. Let
A
and
B
be
n
×
n
matrices over the ﬁeld
Q
of rational numbers.
(i) Deﬁne “
A
and
B
are similar over
Q
”.
(ii) True or False: “If
A
and
B
are similar over the ﬁeld
C
of complex numbers,
then
A
and
B
are also similar over
Q
.” Justify your answer.
1
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View Full Document(iii) Let
M
and
N
be
n
×
n
matrices over the polynomial ring
Q
[
t
]. Deﬁne “
M
and
N
are equivalent over
Q
[
t
]”.
(iv) True or False: “Every matrix
M
∈
Q
[
t
]
n
×
n
is equivalent over
Q
[
t
] to a diagonal
matrix.” Justify your answer.
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 Spring '09
 Linear Algebra, Vector Space, Ring, linear operator

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