Preliminary Exam - Fall 1977

Problem 1   Let

$$A = \left( \begin{array}{cc} 7 & 15 \\ -2 & -4 \end{array} \right).$$

Find a real matrix $B$ such that $B^{-1}AB$ is diagonal.

Problem 2  

1. Using only the axioms for a field $\mbox{\bf {F}}$, prove that a system of m homogeneous linear equations in n unknowns with
   $m<n$ and coefficients in $\mbox{\bf {F}}$ has a nonzero solution.

2. Use Part 1 to show that if $V$ is a vector space over $\mbox{\bf {F}}$ which is spanned by a finite number of elements, then every maximal linearly independent subset of $V$ has the same number of elements.

Problem 3   Let $T$ be an $n\times n$ complex matrix. Show that

$$\lim_{k\to\infty}T^k=0$$

if and only if all the eigenvalues of $T$ have absolute value less than $1$.

Problem 4   Let $P$ be a linear operator on a finite-dimensional vector space over a finite field. Show that if $P$ is invertible, then $P^n=I$ for some positive integer $n$.

Problem 5  

1. Show that the set of all units in a ring with unity form a group under multiplication. (A unit is an element having a two-sided multiplicative inverse.) _ring>unit

2. In the ring $\mbox{$\mathbb{Z}^{}$}_n$ of integers $\bmod n$, show that $k$ is a unit if and only if $k$ and $n$ are relatively prime.

3. Suppose $n = pq$, where $p$ and $q$ are primes. Prove that the number of units in $\mbox{$\mathbb{Z}^{}$}_n$ is $(p-1)(q-1)$.

Problem 6   Let $u:\mbox{$\mathbb{R}^{2}$} \to \mbox{$\mathbb{R}^{}$}$ be the function defined by
$u(x,y)=x^3-3xy^2$. Show that $u$ is harmonic and find $v:\mbox{$\mathbb{R}^{2}$} \to \mbox{$\mathbb{R}^{}$}$ such that the function $f: \mbox{$\mathbb{C}\,^{}$} \to \mbox{$\mathbb{C}\,^{}$}$ defined by

$$f(x+iy)=u(x,y)+iv(x,y)$$

is analytic.

Problem 7   Evaluate

$$\int_{-\infty}^{\infty}\frac{dx}{1+x^{2n}}$$

where $n$ is a positive integer.

Problem 8   Find all solutions of the differential equation

$$\frac{d^2x}{dt^2}-2\frac{dx}{dt}+x=\sin t$$

subject to the condition $x(0)=1$ and $x'(0)=0$.

Problem 9   Let $f:[0,1] \to \mbox{$\mathbb{R}^{}$}$ be continuously differentiable, with $f(0)=0$. Prove that

$$\sup_{0\leq x\leq 1}\vert f(x)\vert \leq \left(\int_0^1\left(f'(x)\right)^2\,dx\right)^{1/2}.$$

Problem 10   Let $f_n:\mbox{$\mathbb{R}^{}$} \to \mbox{$\mathbb{R}^{}$}$ be differentiable for each
$n = 1, 2, \ldots$ with $\vert f_n'(x)\vert\leq 1$ for all n and x. Assume

$$\lim_{n\to\infty}f_n(x) = g(x)$$

for all x. Prove that $g:\mbox{$\mathbb{R}^{}$}\to\mbox{$\mathbb{R}^{}$}$ is continuous.

Problem 11   Show that the differential equation $x'=3x^2$ has no solution such that $x(0)=1$ and $x(t)$ is defined for all real numbers t.

Problem 12   Let $X \subset \mbox{$\mathbb{R}^{}$}$ be a nonempty connected set of real numbers. If every element of $X$ is rational, prove $X$ has only one element.

Problem 13   Consider the following four types of transformations:

$$z \mapsto z+b, \quad z \mapsto 1/z, \quad z \mapsto kz \quad (where \; k\neq 0),$$

$$z \mapsto \frac{az+b}{cz+d} \quad (where \; ad-bc\neq 0).$$

Here, $z$ is a variable complex number and the other letters denote constant complex numbers. Show that each transformation takes circles to either circles or straight lines.

Problem 14   If $a$ and $b$ are complex numbers and $a\neq 0$, the set $a^b$ consists of those complex numbers $c$ having a logarithm of the form $b\alpha$, for some logarithm $\alpha$ of $a$. (That is, $e^{b\alpha}=c$ and $e^{\alpha}=a$ for some complex number $\alpha$.) Describe set $a^b$ when $a=1$ and $b=1/3+i$.

Problem 15   Let $f:\mbox{$\mathbb{R}^{n}$}\to\mbox{$\mathbb{R}^{}$}$ have continuous partial derivatives and satisfy

$$\left\vert\frac{\partial f}{\partial x_j}(x)\right\vert\leq K$$

for all $x=(x_1,\ldots,x_n)$, $j=1,\ldots,n$. Prove that

$$\vert f(x)-f(y)\vert\leq \sqrt{n}K\Vert x-y\Vert$$

(where $\Vert u\Vert=\sqrt{u_1^2+\cdots+u_n^2}\,$).

Problem 16   Let $E$ and $F$ be vector spaces (not assumed to be finite-dimensional). Let $S : E \to F$ be a linear transformation.

1. Prove $S(E)$ is a vector space.

2. Show $S$ has a kernel $\{0\}$ if and only if $S$ is injective (i.e., one-to-one).

3. Assume $S$ is injective; prove $S^{-1}:S(E) \to E$ is linear.

Problem 17   Let $G$ be the set of 3$\times$3 real matrices with zeros below the diagonal and ones on the diagonal.

1. Prove $G$ is a group under matrix multiplication.

2. Determine the center of $G$.

Problem 18   Suppose the complex number $\alpha$ is a root of a polynomial of degree $n$ with rational coefficients. Prove that $1/\alpha$ is also a root of a polynomial of degree $n$ with rational coefficients.

Problem 19   Let $M$ be a real 3$\times$3 matrix such that $M^3 = I$, $M \neq I$.

1. What are the eigenvalues of $M$?

2. Give an example of such a matrix.

Problem 20   Let $\mathbb{C}\,^{3}$ denote the set of ordered triples of complex numbers. Define a map $F:\mbox{$\mathbb{C}\,^{3}$} \to \mbox{$\mathbb{C}\,^{3}$}$ by

$$F(u,v,w) = (u+v+w,uv+vw+wu,uvw).$$

Prove that $F$ is onto but not one-to-one.