Previous: Summer84
Next: Spring85
Preliminary Exam - Fall 1984
Problem 2
Let
![$A$](img10-23.gif)
and
![$B$](img11-23.gif)
be n
![$\times$](img12-23.gif)
n real matrices, and k a positive
integer. Find
-
-
Problem 3
Prove or supply a counterexample: If
![$f$](img15-23.gif)
is a nondecreasing
real valued function on
![$[0,1]$](img16-23.gif)
, then there is a sequence
of continuous functions on
![$[0,1]$](img17-23.gif)
,
![$\{f_n\}$](img18-23.gif)
,
such
that for each
![$x \in [0,1]$](img19-23.gif)
,
Problem 5
Consider the differential equation
Prove
- For each
, there is a unique solution
defined for
such that
.
-
Problem 6
Let
![$\gcd$](img28-23.gif)
abbreviate greatest common divisor and
![$\mathrm{lcm}$](img29-23.gif)
abbreviate
least common multiple. For three nonzero integers a, b, c, show
that
Problem 7
Let
![$[x_1,\ldots,x_n]$](img32-23.gif)
be the polynomial ring
over the real field
![% latex2html id marker 789
$\mathbb{R}^{}$](img33-23.gif)
in the
![$n$](img34-23.gif)
variables
![$x_1,\ldots,x_n$](img35-23.gif)
.
Let the matrix
![$A$](img36-23.gif)
be the
![$n \times n$](img37-23.gif)
matrix whose
![$i^{th}$](img38-23.gif)
row is
![$(1, x_i, x_i^2, \ldots ,x_i^{n-1})$](img39-23.gif)
,
![$i = 1, \ldots , n$](img40-23.gif)
.
Show that
Problem 8
Let
![$a$](img42-23.gif)
,
![$b$](img43-23.gif)
,
![$c$](img44-23.gif)
, and
![$d$](img45-23.gif)
be real numbers, not all zero. Find the
eigenvalues of the following 4
![$\times $](img46-23.gif)
4 matrix and describe
the eigenspace decomposition of
![% latex2html id marker 837
$\mathbb{R}^{4}$](img47-23.gif)
:
Problem 9
Let
![$f$](img49-23.gif)
and
![$g$](img50-23.gif)
be analytic functions in the open unit disc,
and let
![$C_r$](img51-23.gif)
denote the circle with center
![$0$](img52-23.gif)
and radius
![$r$](img53-23.gif)
,
oriented counterclockwise.
- Prove that the integral
is independent of
as long as
and that it
defines an analytic function
,
.
- Prove or supply a counterexample: If
and
, then
.
Problem 10
Prove or supply a counterexample:
- If
and
are
real valued functions on
, if
if
and
never vanish, and if
then
- Do the same question for complex valued
and
.
Problem 11
Show that all groups of order
![$\leq 5$](img73-23.gif)
are commutative. Give an
example of a noncommutative group of order
![$6$](img74-22.gif)
.
Problem 12
Let
![$\theta$](img75-22.gif)
and
![$\varphi$](img76-22.gif)
be fixed,
![$0 \leq \theta \leq 2\pi$](img77-22.gif)
,
![$0 \leq \varphi \leq 2\pi$](img78-22.gif)
and let
![$R$](img79-22.gif)
be the linear transformation
from
![% latex2html id marker 994
$\mathbb{R}^{3}$](img80-22.gif)
to
![% latex2html id marker 996
$\mathbb{R}^{3}$](img81-22.gif)
whose matrix in the standard basis
![$\vec{\imath}$](img82-22.gif)
,
![$\vec{\jmath}$](img83-22.gif)
, and
![$\vec{k}$](img84-22.gif)
is
Let
![$S$](img86-22.gif)
be the linear transformation of
![% latex2html id marker 1006
$\mathbb{R}^{3}$](img87-22.gif)
to
![% latex2html id marker 1008
$\mathbb{R}^{3}$](img88-21.gif)
whose
matrix with respect to the basis
is
Prove that
![$T = R\circ S$](img91-21.gif)
leaves a line invariant.
Problem 13
Show that if
![$f$](img92-21.gif)
is a homeomorphism of
![$[0,1]$](img93-21.gif)
onto itself, then
there is a sequence
![$\{p_n\}$](img94-21.gif)
,
![$n = 1, 2, 3, \ldots\,$](img95-21.gif)
of polynomials
such that
![$p_n \to f$](img96-21.gif)
uniformly on
![$[0,1]$](img97-21.gif)
and each
![$p_n$](img98-21.gif)
is a
homeomorphism of
![$[0,1]$](img99-21.gif)
onto itself.
Problem 15
Consider the differential equation
- Show that
decreases along solutions.
- Show that for any
, there is a
such
that whenever
, there is a
unique solution
of the given equations with the initial
condition
which is defined for all
and
satisfies
.
Problem 16
Let
![$a$](img111-14.gif)
be an element in a field
![% latex2html id marker 1116
$\mbox{\bf {F}}$](img112-14.gif)
and let
![$p$](img113-14.gif)
be a prime. Assume a
is not a
![$p^{th}$](img114-12.gif)
power. Show that the polynomial
![$x^p-a$](img115-12.gif)
is
irreducible in
![% latex2html id marker 1124
$\mbox{\bf {F}}[x]$](img116-11.gif)
.
Problem 17
Let
![$M$](img117-10.gif)
be the
![$n\times n$](img118-9.gif)
matrix over a field
![% latex2html id marker 1144
$\mbox{\bf {F}}$](img119-9.gif)
all of whose
entries are equal to
![$1$](img120-9.gif)
. Find the Jordan Canonical Form of
![$M$](img121-9.gif)
and
discuss the extent to which the Jordan form depends on the
characteristic of the field
![% latex2html id marker 1150
$\mbox{\bf {F}}$](img122-9.gif)
.
Problem 18
Let
![$P_n$](img123-9.gif)
be the vector space of all real polynomials with degrees
at most n. Let
![$D:P_n \to P_n$](img124-8.gif)
be given by differentiation:
![$D(p) = p'$](img125-7.gif)
. Let
![$\pi$](img126-7.gif)
be a real polynomial. What is the
minimal polynomial of the transformation
![$\pi(D)$](img127-7.gif)
?
Problem 19
Prove or supply a counterexample: If
![$f$](img128-7.gif)
is a continuous complex
valued function defined on a connected open subset of the
complex plane and if
![$f^2$](img129-7.gif)
is analytic, then
![$f$](img130-7.gif)
is analytic.
Problem 20
Let
![$f$](img131-7.gif)
be a
![$C^2$](img132-7.gif)
function on the real line. Assume
![$f$](img133-7.gif)
is
bounded with bounded second derivative. Let
Prove that
Previous: Summer84
Next: Spring85
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10