Preliminary Exam - Spring 1986

Problem 1   Let $e = (a, b, c)$ be a unit vector in $\mathbb{R}^{3}$ and let ${T}$ be the linear transformation on $\mathbb{R}^{3}$ of rotation by $180^{\circ}$ about e. Find the matrix for ${T}$ with respect to the standard basis.

Problem 2   Let f be a continuous real valued function on $\mathbb{R}^{}$ such that

$$f(x)=f(x+1)=f\left(x+\sqrt{2}\right)$$

for all x. Prove that f is constant.

Problem 3   Let $C$ be a simple closed contour enclosing the points $0, 1, 2, \ldots, k$ in the complex plane, with positive orientation. Evaluate the integrals

$$I_k = \int_C\frac{dz}{z(z-1)\cdots(z-k)}, \hspace{.2in} k=0,1, \ldots,$$

$$J_k = \int_C\frac{(z-1)\cdots(z-k)}{z}\,dz, \hspace{.2in} k = 0, 1, \ldots .$$

Problem 4   Let $f$ be a positive differentiable function on $(0,\infty)$. Prove that

$$\lim_{\delta \to 0} \left( \frac{f(x+\delta x)} {f(x)} \right)^{1/\delta}$$

exists (finitely) and is nonzero for each x.

Problem 5   Prove that there exists only one automorphism of the field of real numbers; namely the identity automorphism.

Problem 6   Let $V$ be a finite-dimensional vector space and $A$ and $B$ two linear transformations of $V$ into itself such that $A^2=B^2=0$ and
$AB + BA=I$.

1. Prove that if $N_A$ and $N_B$ are the respective null spaces of $A$ and $B,$ then $N_A=AN_B$, $N_B=BN_A$, and $V=N_A \oplus N_B$.
2. Prove that the dimension of $V$ is even.
3. Prove that if the dimension of $V$ is $2$, then $V$ has a basis with respect to which $A$ and $B$ are represented by the matrices
   
   $$\left( \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right)\... ... \left( \begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array} \right).$$
   
   

Problem 7   For $\lambda$ a real number, find all solutions of the integral equations

$$\varphi(x) = e^x + \lambda\int_0^xe^{(x-y)}\varphi(y)\,dy, \hspace{.2in} 0 \leq x \leq 1,$$

$$\psi(x) = e^x + \lambda\int_0^1e^{(x-y)}\psi(y)\,dy, \hspace{.2in} 0 \leq x \leq 1.$$

Problem 8   Let the 3$\times$3 matrix function $A$ be defined on the complex plane by

$$A(z) = \left( \begin{array}{ccc} 4z^2 & 1 & -1 \\ -1 & 2z^2 & 0 \\ 3 & 0 & 1\end{array} \right).$$

How many distinct values of $z$ are there such that $\vert z\vert<1$ and $A(z)$ is not invertible?

Problem 9   Let $\mathbb{Z}^{2}$ be the group of lattice points in the plane (ordered pairs of integers, with coordinatewise addition as the group operation). Let $H_1$ be the subgroup generated by the two elements $(1,2)$ and $(4,1)$, and $H_2$ the subgroup generated by the two elements $(3,2)$ and $(1,3)$. Are the quotient groups $G_1 = \mbox{$\mathbb{Z}^{2}$}/H_1$ and $G_2 = \mbox{$\mathbb{Z}^{2}$}/H_2$ isomorphic?

Problem 10   Suppose addition and multiplication are defined on $\mathbb{C}\,^{n}$, complex $n$-space, coordinatewise, making $\mathbb{C}\,^{n}$ into a ring. Find all ring homomorphisms of $\mathbb{C}\,^{n}$ onto $\mathbb{C}\,^{}$.

Problem 11   Let the complex valued functions $f_n$, $n \in \mbox{$\mathbb{Z}^{}$}$, be defined on $\mathbb{R}^{}$ by _function,>orthonormal

$$f_n(x) = \pi^{-1/2}(x-i)^n/(x+i)^{n+1}.$$

Prove that these functions are orthonormal; that is,

$$% latex2html id marker 939 \int_{-\infty}^{\infty}f_m(x)\ov... ...m{if} & m=n \\ 0 & \mathrm{if} & m\neq n. \end{array} \right.$$

Problem 12   Let $f$ be a real valued continuous function on $\mathbb{R}^{}$ satisfying the mean value inequality below:

$$% latex2html id marker 978 f(x) \leq \frac{1}{2h}\int_{x-h}... ...hspace{.2in} x \in \mbox{$\mathbb{R}^{}$}, \hspace{.2in} h > 0.$$

Prove:

1. The maximum of $f$ on any closed interval is assumed at one of the endpoints.

2. $f$ is convex.

_function,>convex

Problem 13   Let $S$ be a nonempty commuting set of $n\times n$ complex matrices $(n\geq 1)$. Prove that the members of $S$ have a common eigenvector.

Problem 14   Let $K$ be a compact subset of $\mathbb{R}^{n}$ and $\{B_j\}$ a sequence of open balls that covers $K$. Prove that there is a positive number $\varepsilon$ such that each $\varepsilon$-ball centered at a point of $K$ is contained in one of the balls $B_j$.

Problem 15   Consider $\mathbb{R}^{2}$ be equipped with the Euclidean metric
$d(x,y) = \Vert x-y\Vert$. Let $T$ be an isometry of $\mathbb{R}^{2}$ into itself. Prove that $T$ can be represented as $T(x) = a + U(x)$, where a is a vector in $\mathbb{R}^{2}$ and $U$ is an orthogonal linear transformation.

Problem 16   Let $\mathbb{Z}^{}$ be the ring of integers, $p$ a prime, and
$\mbox{\bf {F}}_p = \mbox{$\mathbb{Z}^{}$}/p\mbox{$\mathbb{Z}^{}$}$ the field of $p$ elements. Let $x$ be an indeterminate, and set
$R_1=\mbox{\bf {F}}_p[x]/\langle x^2-2 \rangle$, $R_2 = \mbox{\bf {F}}_p[x]/\langle x^2-3\rangle$. Determine whether the rings $R_1$ and $R_2$ are isomorphic in each of the cases $p = 2, 5, 11$.

Problem 17   Let $V$ be a finite-dimensional vector space (over $\mathbb{C}\,^{}$) of $C^{\infty}$ complex valued functions on $\mathbb{R}^{}$ (the linear operations being defined pointwise). Prove that if $V$ is closed under differentiation (i.e., $f'(x)$ belongs to $V$ whenever $f(x)$ does), then $V$ is closed under translations (i.e., $f(x+a)$ belongs to $V$ whenever $f(x)$ does, for all real numbers $a$).

Problem 18   Let $f$, $g_1$, $g_2,\ldots$ be entire functions. Assume that

1. $\vert g_n^{(k)}(0)\vert \leq \vert f^{(k)}(0)\vert$ for all $n$ and $k$;

2. $${\lim_{n\to\infty}g_n^{(k)}(0)}$$ exists for all $k$.

Prove that the sequence $\{g_n\}$ converges uniformly on compact sets and that its limit is an entire function.

Problem 19   Prove that the additive group of $\mathbb{Q}\,^{}$, the rational number field, is not finitely generated.

Note: See also Problems and .

Problem 20   Evaluate

$$\int_{\vert z\vert=1}(e^{2\pi z}+1)^{-2}\,dz$$

where the integral is taken in counterclockwise direction.