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Preliminary Exam - Fall 1999
Problem 1
Let
![$V$](img1-54.gif)
and
![$W$](img2-54.gif)
be finite dimensional vector spaces, let
![$X$](img3-54.gif)
be a
subspace of
![$W$](img4-54.gif)
, and let
![$T: V \rightarrow W$](img5-54.gif)
be a linear map. Prove
that the dimension of
![$T^{-1}(X)$](img6-54.gif)
is at least
![$\dim V - \dim W + \dim X$](img7-54.gif)
.
Problem 2
Let
![$E_1,E_2,\dots$](img8-54.gif)
be nonempty closed subsets of a complete metric
space
![$(X,d)$](img9-54.gif)
with
![$E_{n+1} \subset E_n$](img10-54.gif)
for all positive integers
![$n$](img11-54.gif)
,
and such that
![$\lim_{n\to\infty} \mathrm{diam}\,(E_n) = 0$](img12-54.gif)
,
where
![$\mathrm{diam}\,(E)$](img13-54.gif)
is defined to be
Prove that
![$\cap_{n=1}^{\infty} E_n \neq \emptyset$](img15-54.gif)
.
Problem 3
Let
![$R$](img16-54.gif)
be a ring with
![$1$](img17-54.gif)
. Suppose that
![$A_1$](img18-54.gif)
,
![$A_2$](img19-54.gif)
,
![$\dots$](img20-54.gif)
,
![$A_n$](img21-54.gif)
are left ideals in
![$R$](img22-54.gif)
such that
![$R = A_1 \oplus
A_2 \oplus \cdots \oplus A_n$](img23-54.gif)
(as additive groups). Prove that there
are elements
![$u_i \in A_i$](img24-54.gif)
such that for any elements
![$a_i \in A_i$](img25-54.gif)
,
![$a_iu_i = a_i$](img26-54.gif)
and
![$a_iu_j = 0$](img27-54.gif)
if
![$j \neq i$](img28-54.gif)
.
Problem 4
Let the rational function
![$f$](img29-54.gif)
in the complex plane have no poles for
![$\Im z \ge 0$](img30-54.gif)
. Prove that
Problem 5
Let
![$M_n$](img32-54.gif)
be the vector space of real n
![$\times$](img33-54.gif)
n matrices, identified
in the usual way with the Euclidean space
![% latex2html id marker 754
$\mbox{$\mathbb{R}^{n^2}$}$](img34-54.gif)
. (Thus, the
norm of a matrix
![$X = (x_{jk})_{j,k=1}^n$](img35-54.gif)
in
![$M_n$](img36-54.gif)
is given by
![$\Vert X\Vert^2
= \sum_{j,k=1}^n x_{jk}^2$](img37-54.gif)
.) Define the map
![$f$](img38-54.gif)
of
![$M_n$](img39-54.gif)
into
![$M_n$](img40-54.gif)
by
![$f(X) = X^2$](img41-54.gif)
. Determine the derivative
![$Df$](img42-54.gif)
of
![$f$](img43-54.gif)
.
Problem 6
Let
![$T: V \to V$](img44-54.gif)
be a linear operator on an
![$n$](img45-54.gif)
dimensional
vector space
![$V$](img46-54.gif)
over a field
![% latex2html id marker 793
$\mbox{\bf {F}}$](img47-54.gif)
. Prove that
![$T$](img48-54.gif)
has an
invariant subspace
![$W$](img49-54.gif)
other than
![$\{0\}$](img50-54.gif)
and
![$V$](img51-54.gif)
if and only if the
characteristic polynomial of
![$T$](img52-54.gif)
has a factor
![% latex2html id marker 805
$f \in \mbox{\bf {F}}[t]$](img53-54.gif)
with
![$0 < \mbox{deg } f < n$](img54-54.gif)
.
Problem 7
Let
![$G$](img55-54.gif)
be a finite group acting transitively on a set
![$X$](img56-54.gif)
of size at
least
![$2$](img57-54.gif)
. Prove that some element
![$g$](img58-54.gif)
of
![$G$](img59-54.gif)
acts without fixed
points.
Problem 8
Evaluate the integral
where the direction of integration is counterclockwise.
Problem 9
Describe all three dimensional vector spaces
![$V$](img61-54.gif)
of
![$C^{\infty}$](img62-54.gif)
complex valued functions on
![% latex2html id marker 853
$\mbox{$\mathbb{R}^{}$}$](img63-54.gif)
that are invariant under the
operator of differentiation.
Problem 10
Let
![$f$](img64-54.gif)
be a continuous real valued function on
![$[0,\infty)$](img65-54.gif)
such that
![$\lim_{x \to\infty} f(x)$](img66-54.gif)
exists (finitely). Prove that
![$f$](img67-54.gif)
is uniformly continuous.
Problem 11
Let
![$V$](img68-54.gif)
be a finite dimensional vector space over a field
![% latex2html id marker 879
$\mbox{\bf {F}}$](img69-54.gif)
,
and let
![$A$](img70-54.gif)
and
![$B$](img71-54.gif)
be diagonalizable linear operators on
![$V$](img72-54.gif)
such that
![$AB = BA$](img73-54.gif)
. Prove that
![$A$](img74-53.gif)
and
![$B$](img75-53.gif)
are simultaneously diagonalizable,
in other words, that there is a basis for
![$V$](img76-53.gif)
consisting of
eigenvectors of both
![$A$](img77-53.gif)
and
![$B$](img78-53.gif)
.
Problem 12
Let
![% latex2html id marker 923
$A = \{0\} \cup \{1/n\;\vert\; n \in \mbox{$\mathbb{Z}^{}$},n > 1\}$](img79-53.gif)
, and let
![% latex2html id marker 927
$\mbox{$\mathbb{D}^{}$}$](img80-53.gif)
be the
open unit disc in the complex plane. Prove that every bounded
holomorphic function on
![% latex2html id marker 931
$\mbox{$\mathbb{D}^{}$}\setminus A$](img81-52.gif)
extends to a holomorphic
function on
![% latex2html id marker 935
$\mbox{$\mathbb{D}^{}$}$](img82-52.gif)
.
Problem 13
Let
![% latex2html id marker 967
$\mbox{\bf {K}}$](img83-51.gif)
be the field
![% latex2html id marker 971
$\mbox{$\mathbb{Q}\,^{}$}(\sqrt[10]{2})$](img84-50.gif)
. Prove that
![% latex2html id marker 973
$\mbox{\bf {K}}$](img85-49.gif)
has
degree
![$10$](img86-49.gif)
over
![% latex2html id marker 979
$\mbox{$\mathbb{Q}\,^{}$}$](img87-47.gif)
, and that the group of automorphisms of
![% latex2html id marker 981
$\mbox{\bf {K}}$](img88-46.gif)
has order
![$2$](img89-46.gif)
.
Problem 14
Show that every infinite closed subset of
![% latex2html id marker 1005
$\mbox{$\mathbb{R}^{2}$}$](img90-46.gif)
is the closure
of a countable set.
Problem 15
Let
![$A$](img91-45.gif)
be an n
![$\times$](img92-44.gif)
n complex matrix such that
![$\mathrm{tr}\, A^k = 0$](img93-44.gif)
for
![$k = 1,\dots,n$](img94-44.gif)
. Prove that
![$A$](img95-42.gif)
is nilpotent.
Problem 16
For
![$0 < a < b$](img96-42.gif)
, evaluate the integral
Problem 17
Show that a group
![$G$](img98-42.gif)
is isomorphic to a subgroup of the additive group
of the rationals if and only if
![$G$](img99-40.gif)
is countable and every finite
subset of
![$G$](img100-39.gif)
is contained in an infinite cyclic subgroup of
![$G$](img101-39.gif)
.
Problem 18
Let
![$f$](img102-39.gif)
and
![$g$](img103-36.gif)
be continuous real valued functions on
![% latex2html id marker 1074
$\mbox{$\mathbb{R}^{}$}$](img104-35.gif)
such
that
![$\lim_{\vert x\vert \to\infty} f(x) = 0$](img105-34.gif)
and
![$\int_{-\infty}^{\infty} \vert g(x)\vert dx < \infty$](img106-31.gif)
. Define the function
![$h$](img107-30.gif)
on
![% latex2html id marker 1084
$\mbox{$\mathbb{R}^{}$}$](img108-29.gif)
by
Prove that
![$\lim_{\vert x\vert \to\infty} h(x) = 0$](img110-29.gif)
.
Previous: Spring99
Next: Spring00
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10