# Preliminary Exam - Summer 1980

Problem 1 Exhibit a real 3×3 matrix having minimal polynomial $( t ^ { 2 } + 1 ) ( t -$ 10), which, as a linear transformation of $\mathbb { R } ^ { 3 }$ , leaves invariant the line L through (0, 0, 0) and (1, 1, 1) and the plane through (0, 0, 0) perpendicular to L.

Problem 2 Which of the following matrix equations have a real matrix solution X? (It is not necessary to exhibit solutions.)

1.

$$
\begin{array}{c} X ^ { 3 } = { \binom { 0 } { 1 } } \ 0 \ 0  \\ { 2 \ 3 \ 0 } \end{array} ) ,
$$

2.

$$
2 X ^ { 5 } + X = { \left( \begin{array} { l l l } { 3 } & { 5 } & { 0 } \\ { 5 } & { 1 } & { 9 } \\ { 0 } & { 9 } & { 0 } \end{array} \right) } \ ,
$$

3.

$$
X ^ { 6 } + 2 X ^ { 4 } + 1 0 X = { \binom { 0 } { 1 } } \quad { \overset { } { 0 } } \quad
$$

4.

$$
X ^ { 4 } = { \binom { 3 \ 4 \quad 0 } { 0 } } \ .
$$

Problem 3 Let $T : V  V$ be an invertible linear transformation of a vector space V . Denote by G the group of all maps $f _ { k , a } : V \to V$ where $k \in \mathbb { Z } , a \in V$ , and for $x \in V$

$$
f _ { k , a } ( x ) = T ^ { k } x + a \quad ( x \in V ) .
$$

Prove that the commutator subgroup $G ^ { \prime }$ of G is isomorphic to the additive group of the vector space $( T - I ) V$ , the image of $T - I$ . (G0 is generated by all $g h g ^ { - 1 } h ^ { - 1 }$ , g and h in G.)

Problem 4 Let G be a finite group and $H \subset G$ a subgroup.

1. Show that the number of subgroups of G of the form $\mathrm { { c } } H \mathrm { { x } ^ { - 1 } }$ for some $x \in G$ is $\leqslant$ the index of H in G.

2. Prove that some element of G is not in any subgroup of the form x $; H x ^ { - 1 }$ $x \in G$

Problem 5 Consider the differential equations

$$
{ \frac { d x } { d t } } = - x + y , \quad { \frac { d y } { d t } } = \log ( 2 0 + x ) - y .
$$

Let $x ( t )$ and $y ( t )$ be a solution defined for all $t \geqslant 0$ with $x ( 0 ) > 0$ and $y ( 0 ) > 0$ . Prove that x(t) and $y ( t )$ are bounded.

Problem 6 Let C denote the positively oriented circle $| z | = 2 , z \in \mathbb { C }$ . Evaluate the integral

$$
\int _ { C } { \sqrt { z ^ { 2 } - 1 } } d z
$$

where the branch of the square root is chosen so that $\sqrt { 2 ^ { 2 } - 1 } > 0$

Problem 7 Exhibit a conformal map from $\{ z \in \mathbb { C } \mid | z | < 1 , \Re z > 0 \}$ onto $\mathbb { D } = \left\{ z \in \mathbb { C } \ | \ | z | < 1 \right\}$

Problem 8 Give an example of a subset of R having uncountably many connected components. Can such a subset be open? Closed?

Problem 9 For each $( a , b , c ) \in \mathbb { R } ^ { 3 }$ , consider the series

$$
\sum _ { n = 3 } ^ { \infty } { \frac { a ^ { n } } { n ^ { b } ( \log n ) ^ { c } } } \cdotp
$$

Determine the values of $( a , b , c )$ for which the series

1. converges absolutely;

2. converges but not absolutely;

3. diverges.

Problem 10 Let $f : \mathbb { R } ^ { n } \to \mathbb { R }$ be a function whose partial derivatives of order $\leqslant 2$ are everywhere defined and continuous.

1. Let $a \in \mathbb { R } ^ { n }$ be a critical point of f (i.e., $\begin{array} { r } { \frac { \partial f } { \partial \boldsymbol { x } _ { j } } ( \boldsymbol { a } ) = 0 , ~ i = 1 , \ldots , n \big ) } \end{array}$ . Prove that a is a local minimum provided the Hessian matrix

$$
\left( { \frac { \partial ^ { 2 } f } { \partial { x _ { i } } \partial { x _ { j } } } } \right)
$$

is positive definite at $x = a$

2. Assume the Hessian matrix is positive definite at all x. Prove that f has, at most, one critical point.

Problem 11 Prove that every finite group is isomorphic to

1. A group of permutations;

2. A group of even permutations.

Problem 12 Let $S \subset \mathbb { R } ^ { 3 }$ denote the ellipsoidal surface defined by

$$
2 x ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z - 1 0 ) ^ { 2 } = 1 .
$$

Let $T \subset \mathbb { R } ^ { 3 }$ be the surface defined by

$$
z = { \frac { 1 } { x ^ { 2 } + y ^ { 2 } + 1 } } \cdotp
$$

Prove that there exist points $p \in S , \ q \in T$ , such that the line pq is perpendicular to S at p and to T at q.

Problem 13 Let J be the ideal in the ring Z[x] (of polynomials with integer coefficients) generated by $x - 5$ and 14. Find $n \in \mathbb { Z }$ such that $0 \leqslant n \leqslant 1 3$ and $( x ^ { 3 } + 2 x + 1 ) ^ { 5 0 } - n \in \mathfrak { J }$

Problem 14 Let A and B be real ${ \mathcal { Q } } \times { \mathcal { Q } }$ matrices such that $A ^ { 2 } = B ^ { 2 } = I$ and $A B + B A = 0$ . Prove there exists a real nonsingular matrix T with

$$
T A T ^ { - 1 } = \left( \begin{array} { c c } { { 1 } } & { { 0 } } \\ { { 0 } } & { { - 1 } } \end{array} \right) T B T ^ { - 1 } = \left( \begin{array} { c c } { { 0 } } & { { 1 } } \\ { { 1 } } & { { 0 } } \end{array} \right) .
$$

Problem 15 Let E be a finite-dimensional vector space over a field F. Suppose $B : E { \times } E {  } \mathbf { F }$ is a bilinear map (not necessarily symmetric). Define subspaces

$$
E _ { 1 } = \{ x \in E \mid B ( x , y ) = 0 \ f o r a l l \ y \in E \} ,
$$

$$
E _ { 2 } = \{ y \in E \mid B ( x , y ) = 0 \ f o r a l l \ x \in E \}
$$

Prove that dim $E _ { 1 } = \dim E _ { 2 }$

Problem 16 Let $\left( a _ { n } \right)$ be a sequence of nonzero real numbers. Prove that the sequence of functions $f _ { n } : \mathbb { R } \to \mathbb { R }$

$$
f _ { n } ( x ) = { \frac { 1 } { a _ { n } } } \sin ( a _ { n } x ) + \cos ( x + a _ { n } )
$$

has a subsequence converging to a continuous function.

Problem 17 Let $f : \mathbb { R }  \mathbb { R }$ be monotonically increasing (perhaps discontinuous). Suppose $0 < f ( 0 )$ and $f ( 1 0 0 ) < 1 0 0$ . Prove $f ( x ) = x$ for some x.

Problem 18 How many zeros does the complex polynomial

$$
3 z ^ { 9 } + 8 z ^ { 6 } + z ^ { 5 } + 2 z ^ { 3 } + 1
$$

have in the annulus $1 < | z | < 2 \ ?$

Problem 19 Let f be a meromorphic function on C which is analytic in a neighborhood of 0. Let its Maclaurin series be

$$
\sum _ { k = 0 } ^ { \infty } a _ { k } z ^ { k }
$$

with all $a _ { k } \geqslant 0$ . Suppose there is a pole of modulus $r > 0$ and no pole has modulus $< r$ . Prove there is a pole at $z = r$

Problem 20 Prove that the initial value problem

$$
\frac { d x } { d t } = 3 x + 8 5 \cos x , \quad x ( 0 ) = 7 7
$$

has a solution x(t) defined for all $t \in \mathbb { R }$