# Preliminary Exam - Fall 1999

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

Problem 2 Let $E _ { 1 } , E _ { 2 } , . . .$ . 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 diam $\left( E _ { n } \right) = 0$ , where diam (E) is defined to be   
n→∞

$$
\operatorname* { s u p } \{ d ( x , y ) \mid x , y \in E \} .
$$

Prove that $\bigcap _ { n = 1 } ^ { \infty } E _ { n } \neq \varnothing$

Problem 3 Let R be a ring with identity element. Suppose that $\mathfrak { I } _ { 1 } , \mathfrak { I } _ { 2 } , . . . ,$ $\Im _ { n }$ are left ideals in R such that $R = \mathfrak { I } _ { 1 } \oplus \mathfrak { I } _ { 2 } \oplus \cdot \cdot \cdot \oplus \mathfrak { I } _ { n }$ (as additive groups). Prove that there are elements $u _ { i } \in \mathfrak { I } _ { i }$ such that for any elements $a _ { i } \in \Im _ { i } ,$ $a _ { i } u _ { i } = a _ { i }$ and $a _ { i } u _ { j } = 0 \ i f j \neq i$

Problem 4 Let the rational function f in the complex plane have no poles $f o r \mathfrak { I z } \geqslant 0$ . Prove that

$$
\operatorname* { s u p } \{ | f ( z ) | \mid \mathbb { S } z \geqslant 0 \} = \operatorname* { s u p } \{ | f ( z ) | \mid \mathbb { S } z = 0 \} .
$$

Problem 5 Let $M _ { n }$ be the vector space of real n×n matrices, identified in the usual way with the Euclidean space $\mathbb { R } ^ { n ^ { 2 } }$ . (Thus, the norm of a matrix $\boldsymbol { X } = ( x _ { j k } ) _ { j , k = 1 } ^ { n }$ in $M _ { n }$ is given by $\begin{array} { r } { \| X \| ^ { 2 } = \sum _ { j , k = 1 } ^ { n } x _ { j k } ^ { 2 } . \quad } \end{array}$ 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 → V be a linear operator on an n dimensional vector space V over a field 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 $f \in \mathbf { F } [ t ]$ with 0 < 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

$$
I = \frac { 1 } { 2 \pi i } \int _ { | z | = 1 } \frac { ( z + 2 ) ^ { 2 } } { z ^ { 2 } ( 2 z - 1 ) } d z ,
$$

where the direction of integration is counterclockwise.

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

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

Problem 11 Let V be a finite dimensional vector space over a field F, and let A and B be diagonalizable linear operators on V such that $A B = B A$ 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 $A = \{ 0 \} \cup \{ 1 / n \mid n \in \mathbb { Z } , n > 1 \}$ , and let D be the open unit disc in the complex plane. Prove that every bounded holomorphic function on D \ A extends to a holomorphic function on D.

Problem 13 Let K be the field $\mathbb { Q } \left( \sqrt [ 1 0 ] { 2 } \right)$ . Prove that K has degree 10 over Q , and that the group of automorphisms of K has order 2.

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

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

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

$$
I = \frac { 1 } { 2 \pi } \int _ { 0 } ^ { 2 \pi } \frac { 1 } { | a e ^ { i \theta } - b | ^ { 4 } } d \theta .
$$

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 R such that $\scriptstyle \operatorname* { l i m } _ { | x | \to \infty } f ( x ) = 0$ and $\textstyle \int _ { - \infty } ^ { \infty } | g ( x ) | d x < \infty$ . Define the function h on R by

$$
h ( x ) = \int _ { - \infty } ^ { \infty } f ( x - y ) g ( y ) d y .
$$

Prove that lim h(x) = 0. |x|→∞