# Preliminary Exam - Spring 1986

Problem 1 Let $\boldsymbol { e } = \left( a , b , c \right)$ be a unit vector in $\mathbb { R } ^ { 3 }$ and let T be the linear transformation on $\mathbb { R } ^ { 3 }$ of rotation by $1 8 0 ^ { \circ }$ about e. Find the matrix for T with respect to the standard basis.

Problem 2 Let f be a continuous real valued function on 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

$$
I _ { k } = \int _ { C } { \frac { d z } { z ( z - 1 ) \cdots ( z - k ) } } , \quad k = 0 , 1 , \ldots ,
$$

$$
J _ { k } = \int _ { C } { \frac { ( z - 1 ) \cdots ( z - k ) } { z } } d z , \quad k = 0 , 1 , \ldots .
$$

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

$$
\operatorname* { l i m } _ { \delta  0 } ( \frac { f ( x + \delta x ) } { f ( x ) } ) ^ { 1 / \delta }
$$

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 $A ^ { 2 } = B ^ { 2 } = 0$ andAB+BA = I .

1. Prove that if $N _ { A }$ and $N _ { B }$ are the respective null spaces of A and B, then $N _ { A } = A N _ { B } , N _ { B } = B N _ { A }$ , 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

$$
\left( \begin{array} { c c } { { 0 } } & { { 1 } } \\ { { 0 } } & { { 0 } } \end{array} \right) \quad a n d \quad \left( \begin{array} { c c } { { 0 } } & { { 0 } } \\ { { 1 } } & { { 0 } } \end{array} \right) .
$$

Problem 7 For λ a real number, find all solutions of the integral equations

$$
\varphi ( x ) = e ^ { x } + \lambda \int _ { 0 } ^ { x } e ^ { ( x - y ) } \varphi ( y ) d y , \quad 0 \leqslant x \leqslant 1 ,
$$

$$
\psi ( x ) = e ^ { x } + \lambda \int _ { 0 } ^ { 1 } e ^ { ( x - y ) } \psi ( y ) d y , \quad 0 \leqslant x \leqslant 1 .
$$

Problem 8 Let the $\mathcal { B } { \times } \mathcal { B }$ matrix function A be defined on the complex plane by

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

How many distinct values of z are there such that $| z | < 1$ 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 $G _ { 1 } = \mathbb { Z } ^ { 2 } / H _ { 1 }$ and $G _ { 2 } = \mathbb { Z } ^ { 2 } / H _ { 2 }$ 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 C .

Problem 11 Let the complex valued functions $f _ { n } , n \in \mathbb { Z }$ , be defined on R by

$$
f _ { n } ( x ) = { \frac { ( x - i ) ^ { n } } { \sqrt { \pi } ( x + i ) ^ { n + 1 } } } \ .
$$

Prove that these functions are orthonormal; that is,

$$
\int _ { - \infty } ^ { \infty } f _ { m } ( x ) { \overline { { f _ { n } ( x ) } } } d x = { \left\{ \begin{array} { l l } { 1 } & { { \mathrm { i f } } \quad m = n } \\ { 0 } & { { \mathrm { i f } } \quad m \neq n . } \end{array} \right. }
$$

Problem 12 Let f be a real valued continuous function on R satisfying the mean value inequality below:

$$
f ( x ) \leqslant { \frac { 1 } { 2 h } } \int _ { x - h } ^ { x + h } f ( y ) d y , \quad x \in \mathbb { R } , \quad h > 0 .
$$

Prove:

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

2. f is convex.

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

Problem 14 Let K be a compact subset $o f \mathbb { R } ^ { n }$ and $\{ B _ { j } \}$ a sequence of open balls that covers K. Prove that there is a positive number ε such that each ε-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 $l ( x , y ) =$ $\| x - y \|$ . 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 Z be the ring of integers, p a prime, andF $\mathbf { \Phi } _ { p } ^ { 1 } = \mathbb { Z } / p \mathbb { Z }$ the field of p elements. Let x be an indeterminate, and se $t R _ { 1 } = { \bf F } _ { p } [ x ] / \langle x ^ { 2 } - 2 \rangle$ $R _ { 2 } = { \bf F } _ { p } [ x ] / \langle x ^ { 2 } - 3 \rangle$ . Determine whether the rings $R _ { 1 }$ and $R _ { 2 }$ are isomorphic in each of the cases $p = 2 , 5 , 1 1$

Problem 17 Let V be a finite-dimensional vector space (over C ) of $C ^ { \infty }$ complex valued functions on R (the linear operations being defined pointwise). Prove that if V is closed under differentiation $( i . e . , \ f ^ { \prime } ( x )$ 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. $| g _ { n } ^ { ( k ) } ( 0 ) | \leqslant | f ^ { ( k ) } ( 0 ) |$ for all n and k;

2. $\operatorname* { l i m } _ { n \to \infty } g _ { n } ^ { ( k ) } ( 0 )$ exists for all k.

Prove that the sequence $\left\{ g _ { n } \right\}$ converges uniformly on compact sets and that its limit is an entire function.

Problem 19 Prove that the additive group of Q , the rational number field, is not finitely generated.

Note: See also Problems ?? and ??.

Problem 20 Evaluate

$$
\int _ { | z | = 1 } ( e ^ { 2 \pi z } + 1 ) ^ { - 2 } d z
$$

where the integral is taken in counterclockwise direction.