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Preliminary Exam - Fall 1983
Problem 1
Evaluate
where
![$\lambda$](img2-21.gif)
is a real constant and
Problem 2
Let
![% latex2html id marker 686
$M_{n\times n}(\mbox{\bf {F}})$](img4-21.gif)
denote the ring of n
![$\times$](img5-21.gif)
n
matrices over a field
![% latex2html id marker 690
$\mbox{\bf {F}}$](img6-21.gif)
. For
![$n\geq 1$](img7-21.gif)
does there exist a ring homomorphism from
![% latex2html id marker 694
$M_{(n+1)\times (n+1)}(\mbox{\bf {F}})$](img8-21.gif)
onto
![% latex2html id marker 696
$M_{n\times n}(\mbox{\bf {F}})$](img9-21.gif)
?
Problem 3
Let
![% latex2html id marker 730
$f:\mbox{$\mathbb{R}^{n}$}\setminus \{0\} \to \mbox{$\mathbb{R}^{}$}$](img10-21.gif)
be a function which is continuously
differentiable and whose partial derivatives are uniformly bounded:
for all
![$(x_1,\ldots,x_n)\neq (0,\ldots,0)$](img12-21.gif)
. Show that if
![$n\geq 2$](img13-21.gif)
, then
![$f$](img14-21.gif)
can be extended to a continuous function defined on all of
![% latex2html id marker 738
$\mathbb{R}^{n}$](img15-21.gif)
. Show
that this is false if
![$n=1$](img16-21.gif)
by giving a counterexample.
Problem 4
Prove or disprove (by giving a counterexample), the following
assertion: Every infinite sequence
![$x_1, x_2, \ldots $](img17-21.gif)
of real numbers
has either a nondecreasing subsequence or a nonincreasing subsequence.
Problem 5
Let
![$A$](img18-21.gif)
be the
![$n \times n$](img19-21.gif)
matrix which has zeros on the main diagonal
and ones everywhere else. Find the eigenvalues and eigenspaces of
![$A$](img20-21.gif)
and
compute
![$\det(A)$](img21-21.gif)
.
Problem 6
Consider the polynomial
How many zeros (counting multiplicities) does
![$p$](img23-21.gif)
have in the annular
region
![$1 < \vert z\vert < 2$](img24-21.gif)
?
Problem 7
Let
![$G$](img25-21.gif)
be a finite group and
![$G_1 = G \times G$](img26-21.gif)
. Suppose that
![$G_1$](img27-21.gif)
has exactly four normal subgroups. Show that
![$G$](img28-21.gif)
is simple and nonabelian.
Problem 8
Let
![$A$](img29-21.gif)
be a linear transformation on
![% latex2html id marker 813
$\mathbb{R}^{3}$](img30-21.gif)
whose matrix (relative to
the usual basis for
![% latex2html id marker 815
$\mathbb{R}^{3}$](img31-21.gif)
) is both symmetric and orthogonal. Prove
that
![$A$](img32-21.gif)
is either plus or minus the identity, or a rotation by
![$180^{\circ}$](img33-21.gif)
about some axis in
![% latex2html id marker 821
$\mathbb{R}^{3}$](img34-21.gif)
, or a reflection about some
two-dimensional subspace of
![% latex2html id marker 823
$\mathbb{R}^{3}$](img35-21.gif)
.
Problem 9
For which real values of
![$p$](img36-21.gif)
does the differential equation
admit solutions
![$y=f(x)$](img38-21.gif)
with infinitely many critical points?
Problem 10
Let
![% latex2html id marker 859
$f:[0,\infty) \to \mbox{$\mathbb{R}^{}$}$](img39-21.gif)
be a uniformly continuous function with
the property that
exists (as a finite limit). Show that
Problem 11
Prove or supply a counterexample:
If
![$f$](img42-21.gif)
and
![$g$](img43-21.gif)
are
![$C^1$](img44-21.gif)
real valued functions on
![$(0,1)$](img45-21.gif)
, if
if
![$g$](img47-21.gif)
and
![$g'$](img48-21.gif)
never vanish, and if
then
Problem 12
Let
![$r_1,r_2,\ldots,r_n$](img51-21.gif)
be distinct complex numbers. Show that a
rational function of the form
can be written as a sum
for suitable constants
![$A_1,\ldots,A_n$](img54-21.gif)
.
Problem 14
Let
![$V$](img72-21.gif)
be a finite-dimensional complex vector space and let
![$A$](img73-21.gif)
and
![$B$](img74-20.gif)
be
linear operators on
![$V$](img75-20.gif)
such that
![$AB = BA$](img76-20.gif)
. Prove that if
![$A$](img77-20.gif)
and
![$B$](img78-20.gif)
can
each be diagonalized, then there is a basis for
![$V$](img79-20.gif)
which simultaneously
diagonalizes
![$A$](img80-20.gif)
and
![$B$](img81-20.gif)
.
Problem 16
Let
![$F(t) = \left(f_{ij}(t)\right)$](img86-20.gif)
be an n
![$\times$](img87-20.gif)
n matrix of continuously
differentiable functions
![% latex2html id marker 1071
$f_{ij}:\mbox{$\mathbb{R}^{}$} \to \mbox{$\mathbb{R}^{}$}$](img88-19.gif)
, and let
Show that u is differentiable and
Problem 17
Prove that every finite integral domain is a field.
Problem 18
Let
![% latex2html id marker 1105
$f,g:\mbox{$\mathbb{R}^{}$}\to\mbox{$\mathbb{R}^{}$}$](img91-19.gif)
be smooth functions with
![$f(0)=0$](img92-19.gif)
and
![$f'(0)\neq 0$](img93-19.gif)
. Consider the equation
![$f(x)=tg(x)$](img94-19.gif)
,
![% latex2html id marker 1115
$t\in\mbox{$\mathbb{R}^{}$}$](img95-19.gif)
.
- Show that in a suitably small interval
, there is
a unique continuous function
which solves the equation and
satisfies
.
- Derive the first order Taylor expansion of
about
.
Problem 19
Prove that if
![$p$](img101-18.gif)
is a prime number, then the polynomial
is irreducible in
![% latex2html id marker 1148
$\mathbb{Q}[x]$](img103-15.gif)
.
Problem 20
Let m and n be positive integers, with
![$m<n$](img104-15.gif)
. Let
![$M_{m\times n}$](img105-14.gif)
be the
space of linear transformations of
![% latex2html id marker 1176
$\mathbb{R}^{m}$](img106-13.gif)
into
![% latex2html id marker 1178
$\mathbb{R}^{n}$](img107-13.gif)
(considered as
![$n\times m$](img108-13.gif)
matrices) and let
![$L$](img109-13.gif)
be the set of transformations in
![$M_{m\times n}$](img110-13.gif)
which have rank m.
- Show that
is an open subset of
.
- Show that there is a continuous function
such
that
for all
, where
is the identity on
.
Previous: Summer83
Next: Spring84
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10