Preliminary Exam - Fall 1999

Problem 1   Let $V$ and $W$ be finite dimensional vector spaces, let $X$ be a subspace of $W$, and let $T: V \rightarrow W$ be a linear map. Prove that the dimension of $T^{-1}(X)$ is at least $\dim V - \dim W + \dim X$.

Problem 2   Let $E_1,E_2,\dots$ be nonempty closed subsets of a complete metric space $(X,d)$ with $E_{n+1} \subset E_n$ for all positive integers $n$, and such that $\lim_{n\to\infty} \mathrm{diam}\,(E_n) = 0$, where $\mathrm{diam}\,(E)$ is defined to be

$$\sup\{d(x,y)\;\vert\; x,y \in E\}.$$

Prove that $\cap_{n=1}^{\infty} E_n \neq \emptyset$.

Problem 3   Let $R$ be a ring with $1$. Suppose that $A_1$, $A_2$, $\dots$, $A_n$ are left ideals in $R$ such that $R = A_1 \oplus A_2 \oplus \cdots \oplus A_n$ (as additive groups). Prove that there are elements $u_i \in A_i$ such that for any elements $a_i \in A_i$, $a_iu_i = a_i$ and $a_iu_j = 0$ if $j \neq i$.

Problem 4   Let the rational function $f$ in the complex plane have no poles for $\Im z \ge 0$. Prove that

$$\sup\{\vert f(z)\vert\;\vert\; \Im z \ge 0\} = \sup\{\vert f(z)\vert\;\vert\; \Im z = 0\}.$$

Problem 5   Let $M_n$ be the vector space of real n$\times$n matrices, identified in the usual way with the Euclidean space $\mbox{$\mathbb{R}^{n^2}$}$. (Thus, the norm of a matrix $X = (x_{jk})_{j,k=1}^n$ in $M_n$ is given by $\Vert X\Vert^2 = \sum_{j,k=1}^n x_{jk}^2$.) Define the map $f$ of $M_n$ into $M_n$ by $f(X) = X^2$. Determine the derivative $Df$ of $f$.

Problem 6   Let $T: V \to V$ be a linear operator on an $n$ dimensional vector space $V$ over a field $\mbox{\bf {F}}$. Prove that $T$ has an invariant subspace $W$ other than $\{0\}$ and $V$ if and only if the characteristic polynomial of $T$ has a factor $f \in \mbox{\bf {F}}[t]$ with $0 < \mbox{deg } f < n$.

Problem 7   Let $G$ be a finite group acting transitively on a set $X$ of size at least $2$. Prove that some element $g$ of $G$ acts without fixed points.

Problem 8   Evaluate the integral

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

where the direction of integration is counterclockwise.

Problem 9   Describe all three dimensional vector spaces $V$ of $C^{\infty}$ complex valued functions on $\mbox{$\mathbb{R}^{}$}$ that are invariant under the operator of differentiation.

Problem 10   Let $f$ be a continuous real valued function on $[0,\infty)$ such that $\lim_{x \to\infty} f(x)$ exists (finitely). Prove that $f$ is uniformly continuous.

Problem 11   Let $V$ be a finite dimensional vector space over a field $\mbox{\bf {F}}$, and let $A$ and $B$ be diagonalizable linear operators on $V$ such that $AB = BA$. Prove that $A$ and $B$ are simultaneously diagonalizable, in other words, that there is a basis for $V$ consisting of eigenvectors of both $A$ and $B$.

Problem 12   Let $A = \{0\} \cup \{1/n\;\vert\; n \in \mbox{$\mathbb{Z}^{}$},n > 1\}$, and let $\mbox{$\mathbb{D}^{}$}$ be the open unit disc in the complex plane. Prove that every bounded holomorphic function on $\mbox{$\mathbb{D}^{}$}\setminus A$ extends to a holomorphic function on $\mbox{$\mathbb{D}^{}$}$.

Problem 13   Let $\mbox{\bf {K}}$ be the field $\mbox{$\mathbb{Q}\,^{}$}(\sqrt[10]{2})$. Prove that $\mbox{\bf {K}}$ has degree $10$ over $\mbox{$\mathbb{Q}\,^{}$}$, and that the group of automorphisms of $\mbox{\bf {K}}$ has order $2$.

Problem 14   Show that every infinite closed subset of $\mbox{$\mathbb{R}^{2}$}$ is the closure of a countable set.

Problem 15   Let $A$ be an n$\times$n complex matrix such that $\mathrm{tr}\, A^k = 0$ for $k = 1,\dots,n$. Prove that $A$ is nilpotent.

Problem 16   For $0 < a < b$, evaluate the integral

$$I = \frac {1}{2\pi} \int_0^{2\pi} \frac {1}{\vert ae^{i\theta}-b\vert^4} d\theta\,\cdot$$

Problem 17   Show that a group $G$ is isomorphic to a subgroup of the additive group of the rationals if and only if $G$ is countable and every finite subset of $G$ is contained in an infinite cyclic subgroup of $G$.

Problem 18   Let $f$ and $g$ be continuous real valued functions on $\mbox{$\mathbb{R}^{}$}$ such that $\lim_{\vert x\vert \to\infty} f(x) = 0$ and $\int_{-\infty}^{\infty} \vert g(x)\vert dx < \infty$. Define the function $h$ on $\mbox{$\mathbb{R}^{}$}$ by

$$h(x) = \int_{-\infty}^{\infty} f(x-y)g(y)dy\,.$$

Prove that $\lim_{\vert x\vert \to\infty} h(x) = 0$.