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Preliminary Exam - Summer 1979
Problem 1
Prove that the matrix
has two positive and two negative eigenvalues (counting
multiplicities).
Problem 2
Let
![% latex2html id marker 682
$\mbox{\bf {F}}$](img2-7.gif)
be a subfield of a field
![% latex2html id marker 684
$\mbox{\bf {K}}$](img3-7.gif)
. Let
![$p$](img4-7.gif)
and
![$q$](img5-7.gif)
be
polynomials over
![% latex2html id marker 690
$\mbox{\bf {F}}$](img6-7.gif)
. Prove that their
![$\gcd$](img7-7.gif)
(greatest common
divisor) in the ring of polynomials over
![% latex2html id marker 694
$\mbox{\bf {F}}$](img8-7.gif)
is the same as their
![$\gcd$](img9-7.gif)
in the ring of polynomials over
![% latex2html id marker 698
$\mbox{\bf {K}}$](img10-7.gif)
.
Problem 3
Let
![$X$](img11-7.gif)
be the space of orthogonal real
![$n \times n$](img12-7.gif)
matrices. Let
![% latex2html id marker 733
$v_0 \in \mbox{$\mathbb{R}^{n}$}$](img13-7.gif)
. Locate and describe the elements of
![$X$](img14-7.gif)
, where the map
takes its maximum and minimum values.
Problem 4
Prove that the group of automorphisms of a cyclic group of
prime order
![$p$](img16-7.gif)
is cyclic and find its order.
Problem 6
Let
![$f(z) = a_0 + a_1z + \cdots + a_nz^n$](img23-7.gif)
be a complex polynomial
of degree
![$n > 0$](img24-7.gif)
. Prove
Problem 7
Let
![$f$](img26-7.gif)
be a continuous complex valued function on
![$[0,1]$](img27-7.gif)
, and
define the function
![$g$](img28-7.gif)
by
Prove that
![$g$](img30-7.gif)
is analytic in the entire complex plane.
Problem 8
Let
![% latex2html id marker 851
$U\subset \mbox{$\mathbb{R}^{n}$}$](img31-7.gif)
be an open set. Suppose that the map
![% latex2html id marker 855
$h:U \to \mbox{$\mathbb{R}^{n}$}$](img32-7.gif)
is a homeomorphism from
![$U$](img33-7.gif)
onto
![% latex2html id marker 859
$\mathbb{R}^{n}$](img34-7.gif)
, which is
uniformly continuous. Prove
![% latex2html id marker 863
$U = \mbox{$\mathbb{R}^{n}$}$](img35-7.gif)
.
Problem 9
Prove that a linear transformation
![% latex2html id marker 893
$T : \mbox{$\mathbb{R}^{3}$} \to \mbox{$\mathbb{R}^{3}$}$](img36-7.gif)
has
- a one-dimensional invariant subspace, and
- a two-dimensional invariant subspace.
Problem 10
Find real valued functions of a real variable,
![$x(t)$](img37-7.gif)
,
![$y(t)$](img38-7.gif)
, and
![$z(t)$](img39-7.gif)
, such that
and
Problem 11
Let
![$A$](img42-7.gif)
and
![$B$](img43-7.gif)
be
![$n\times n$](img44-7.gif)
matrices over a field
![% latex2html id marker 936
$\mbox{\bf {F}}$](img45-7.gif)
such that
![$A^2=A$](img46-7.gif)
and
![$B^2=B$](img47-7.gif)
. Suppose that
![$A$](img48-7.gif)
and
![$B$](img49-7.gif)
have the same
rank. Prove that
![$A$](img50-7.gif)
and
![$B$](img51-7.gif)
are similar.
Problem 12
Which rational numbers
![$t$](img52-7.gif)
are such that
is an integer?
Problem 14
Let
![$A$](img68-7.gif)
,
![$B$](img69-7.gif)
, and
![$C$](img70-7.gif)
be finite abelian groups such that A
![$\times$](img71-7.gif)
B and
A
![$\times$](img72-7.gif)
C are isomorphic. Prove that
![$B$](img73-7.gif)
and
![$C$](img74-7.gif)
are isomorphic.
Problem 15
Show that
converges for all complex numbers
![$z$](img76-7.gif)
exterior to the
lemniscate
Problem 16
Let
![$g_n(z)$](img78-7.gif)
be an entire function having only real zeros,
![$n = 1, 2, \ldots$](img79-7.gif)
. Suppose
uniformly on compact sets in
![% latex2html id marker 1084
$\mathbb{C}\,^{}$](img81-7.gif)
, with
![$g$](img82-7.gif)
not identically zero.
Prove that
![$g(z)$](img83-7.gif)
has only real zeros.
Problem 17
Let
![% latex2html id marker 1127
$f:\mbox{$\mathbb{R}^{3}$}\to\mbox{$\mathbb{R}^{}$}$](img84-7.gif)
be such that
Suppose
![$f$](img86-7.gif)
has continuous partial derivatives of orders
![$\leq 2$](img87-7.gif)
.
Is there a
![% latex2html id marker 1137
$y \in \mbox{$\mathbb{R}^{3}$}$](img88-6.gif)
with
![$\Vert y\Vert \leq 1$](img89-6.gif)
such that
Problem 18
Let
![$E$](img91-6.gif)
be a three-dimensional vector space over
![% latex2html id marker 1159
$\mathbb{Q}\,^{}$](img92-6.gif)
. Suppose
![$T:E \to E$](img93-6.gif)
is a linear transformation and
![$Tx=y$](img94-6.gif)
,
![$Ty=z$](img95-6.gif)
,
![$Tz=x+y$](img96-6.gif)
, for certain
![$x, y, z \in E$](img97-6.gif)
,
![$x \neq 0$](img98-6.gif)
. Prove that
![$x$](img99-6.gif)
,
![$y$](img100-6.gif)
, and
![$z$](img101-6.gif)
are linearly independent.
Problem 19
Let
![$\{f_n\}$](img102-6.gif)
be a sequence of continuous functions
![% latex2html id marker 1208
$[0,1] \to \mbox{$\mathbb{R}^{}$}$](img103-4.gif)
such that
for all n.
Define
![% latex2html id marker 1212
$g_n:[0,1] \to\mbox{$\mathbb{R}^{}$}$](img105-3.gif)
by
- Find a constant
such that
for all n.
- Prove that a subsequence of the sequence {
} converges
uniformly.
Problem 20
Let
![% latex2html id marker 1258
$x:\mbox{$\mathbb{R}^{}$}\to\mbox{$\mathbb{R}^{}$}$](img110-3.gif)
be a solution to the differential equation
Prove that the function
![% latex2html id marker 1264
$f:\mbox{$\mathbb{R}^{}$}\to\mbox{$\mathbb{R}^{}$}$](img112-3.gif)
,
attains a maximum value.
Previous: Spring79
Next: Fall79
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10