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Preliminary Exam - Spring 1981
Problem 1
Let
![$\vec{\imath}$](img1-12.gif)
,
![$\vec{\jmath}$](img2-12.gif)
, and
![$\vec{k}$](img3-12.gif)
be the usual unit vectors in
![% latex2html id marker 679
$\mathbb{R}^{3}$](img4-12.gif)
. Let
![$\vec{F}$](img5-12.gif)
denote the vector field
- Compute
(the curl of
).
- Compute the integral of
over
the surface
,
.
Problem 2
Let
![$T$](img12-12.gif)
be a linear transformation of a vector space
![$V$](img13-12.gif)
into itself. Suppose
![$x\in V$](img14-12.gif)
is such that
![$T^m x = 0$](img15-12.gif)
,
![$T^{m-1}x\neq0$](img16-12.gif)
for some positive
integer
![$m$](img17-12.gif)
. Show that
![$x,\, Tx,\ldots , T^{m-1}x$](img18-12.gif)
are linearly independent.
Problem 3
Let
![$D$](img19-12.gif)
be an ordered integral domain and
![$a \in D$](img20-12.gif)
. Prove that
Problem 5
Decompose
![$x^4-4$](img33-12.gif)
and
![$x^3-2$](img34-12.gif)
into irreducibles over
![% latex2html id marker 791
$\mathbb{R}^{}$](img35-12.gif)
, over
![% latex2html id marker 793
$\mathbb{Z}^{}$](img36-12.gif)
, and
over
![% latex2html id marker 797
$\mbox{$\mathbb{Z}^{}$}_3$](img37-12.gif)
(the integers modulo
![$3$](img38-12.gif)
).
Problem 6
Suppose the complex polynomial
has
![$n$](img40-12.gif)
distinct roots
![% latex2html id marker 825
$r_1,\ldots,r_n \in \mbox{$\mathbb{C}\,^{}$}$](img41-12.gif)
. Prove that if
![$\vert b_k-a_k\vert$](img42-12.gif)
is sufficiently small then
has
![$n$](img44-12.gif)
roots which are smooth functions of
![$b_0,\ldots,b_n$](img45-12.gif)
.
Problem 8
Let
![% latex2html id marker 858
$f:[0,1] \to \mbox{$\mathbb{R}^{}$}$](img47-12.gif)
be continuous. Prove that there is a real
polynomial
![$P(x)$](img48-12.gif)
of degree
![$\leq 10$](img49-12.gif)
which minimizes (for all such
polynomials)
Problem 9
Show that the following three conditions are all equivalent for a real
3
![$\times $](img51-12.gif)
3 symmetric matrix
![$A$](img52-12.gif)
, whose eigenvalues are
![$\lambda_1$](img53-12.gif)
,
![$\lambda_2$](img54-12.gif)
, and
![$\lambda_3$](img55-12.gif)
:
-
is not an eigenvalue of
.
-
.
- The map
is an isomorphism, where
is the space of
3
3 real skew-symmetric matrices and
.
Problem 11
Evaluate
where
![$C$](img77-12.gif)
is the closed curve shown below:
file=../Fig/Pr/Sp81-11,width=4.5in
Problem 12
For
![% latex2html id marker 989
$x \in \mbox{$\mathbb{R}^{}$}$](img78-12.gif)
, let
- Prove that
.
- Prove that if
, then
.
Problem 13
Which of the following series converges?
-
-
Problem 14
The set of real 3
![$\times$](img85-12.gif)
3 symmetric matrices is a real, finite-dimensional
vector space isomorphic to
![% latex2html id marker 1038
$\mathbb{R}^{6}$](img86-12.gif)
. Show that the subset of such matrices of
signature
![$(2,1)$](img87-12.gif)
is an open connected subspace in the usual topology on
![% latex2html id marker 1042
$\mathbb{R}^{6}$](img88-11.gif)
.
Problem 15
Let
![% latex2html id marker 1082
$\mbox{\bf {M}}$](img89-11.gif)
be one of the following fields:
![% latex2html id marker 1084
$\mathbb{R}^{}$](img90-11.gif)
,
![% latex2html id marker 1086
$\mathbb{C}\,^{}$](img91-11.gif)
,
![% latex2html id marker 1088
$\mathbb{Q}\,^{}$](img92-11.gif)
, and
![% latex2html id marker 1090
$\mbox{\bf {F}}_{9}$](img93-11.gif)
(the field with nine elements).
Let
![% latex2html id marker 1092
$\mathfrak{I} \subset \mbox{\bf {M}}[x]$](img94-11.gif)
be the ideal
generated by
![$x^4+2x-2$](img95-11.gif)
. For which choices of
![% latex2html id marker 1096
$\mbox{\bf {M}}$](img96-11.gif)
is the
ring
![% latex2html id marker 1098
$\mbox{\bf {M}}[x]/\mathfrak{I}$](img97-11.gif)
a field?
Problem 16
Let
![$f(x)$](img98-11.gif)
be a real valued function defined for all
![$x\geq 1$](img99-11.gif)
, satisfying
![$f(1)=1$](img100-11.gif)
and
Prove that
exists and is less than
Problem 17
Let b be a real nonzero
![$n\times 1$](img104-9.gif)
matrix (a column vector). Set
![$M=bb^t$](img105-8.gif)
(an
![$n\times n$](img106-7.gif)
matrix) where
![$b^t$](img107-7.gif)
denotes the transpose of b.
- Prove that there is an orthogonal matrix
such that
is diagonal, and find
.
- Describe geometrically the linear transformation
.
Problem 18
Describe the two regions in
![$(a,b)$](img112-7.gif)
-space for which the function
restricted to the circle
![$x^2+y^2=1$](img114-5.gif)
, has exactly two, and exactly four
critical points, respectively.
Problem 19
Let
![$G$](img115-5.gif)
be a finite group. A conjugacy class is a set of the form
for some
![$a \in G$](img117-4.gif)
.
- Prove that the number of elements in a conjugacy class divides the
order of
.
- Do all conjugacy classes have the same number of elements?
- If
has only two conjugacy classes, prove
has order
.
Problem 20
Let
![% latex2html id marker 1242
$f:[0,1] \to \mbox{$\mathbb{R}^{}$}$](img122-4.gif)
be continuous with
![$f(0)=0$](img123-4.gif)
. Show there is a
continuous concave function
![% latex2html id marker 1248
$g:[0,1] \to \mbox{$\mathbb{R}^{}$}$](img124-3.gif)
such that
![$g(0)=0$](img125-3.gif)
and
![$g(x)\geq f(x)$](img126-3.gif)
for all
![$x \in [0,1]$](img127-3.gif)
.
Note: A function
is concave if
for all
![$x$](img130-3.gif)
and
![$y$](img131-3.gif)
in
![$I$](img132-3.gif)
and
![$0\leq t \leq 1$](img133-3.gif)
.
_function,>concave
Previous: Fall80
Next: Summer81
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10