# Preliminary Exam - Spring 1979

Problem 1 Let $f : \mathbb { R } ^ { n } \backslash \{ 0 \}  \mathbb { R }$ be differentiable. Suppose

$$
\operatorname* { l i m } _ { x \to 0 } { \frac { \partial f } { \partial x _ { j } } } ( x )
$$

exists for each $j = 1 , \dotsc , n$

1. Can f be extended to a continuous map from $\mathbb { R } ^ { n }$ to R?

2. Assuming continuity at the origin, is f differentiable from $\mathbb { R } ^ { n }$ to R?

Problem 2 Let E denote a finite-dimensional complex vector space with a Hermitian inner product $\langle x , y \rangle$

1. Prove that E has an orthonormal basis.

2. Let $f : E \to \mathbb { C }$ be such that $f ( x , y )$ is linear in x and conjugate linear in $y .$ Show there is a linear map $A : E  E$ such that $f ( x , y ) = \langle A x , y \rangle$

Problem 3 Let $S _ { 7 }$ be the group of permutations of a set of seven objects.   
Find all n such that some element of $S _ { 7 }$ has order $n .$ .

Problem 4 Prove that every compact metric space has a countable dense subset.

Problem 5 Find all solutions to the differential equation

$$
{ \frac { d y } { d x } } = \sqrt y , \qquad y ( 0 ) = 0 .
$$

Problem 6 Prove that if $1 < \lambda < \infty$ , the function

$$
f _ { \lambda } ( z ) = z + \lambda - e ^ { z }
$$

has only one zero in the half-plane $\Re z < 0$ , and that this zero is real.

Problem 7 Evaluate

$$
\int _ { 0 } ^ { \infty } { \frac { x ^ { 2 } + 1 } { x ^ { 4 } + 1 } } d x .
$$

Problem 8 Let M be a real nonsingular $\mathcal { B } { \times } \mathcal { B }$ matrix. Prove there are real matrices S and U such that $M = S U = U S$ , all the eigenvalues of U equal 1, and S is diagonalizable over $\mathbb { C }$

Problem 9 Let M be an $n \times n$ complex matrix. Let $G _ { M }$ be the set of complex numbers λ such that the matrix λM is similar to M .

1. What is $G _ { M }$ if

$$
M = \left( { \begin{array} { c c c } { 0 } & { 0 } & { 4 } \\ { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } \end{array} } \right) ?
$$

2. Assume M is not nilpotent. Prove $G _ { M }$ is finite.

Problem 10 Let $f ( x )$ be a polynomial over $\mathbb { Z } _ { p }$ , the field of integers modp. Let $g ( x ) = x ^ { p } - x$ . Show that the greatest common divisor of $f ( x )$ and $g ( x )$ is the product of the distinct linear factors of $f ( x )$

Problem 11 Classify all abelian groups of order 80 up to isomorphism.

Problem 12 Let G be a group with three normal subgroups $N _ { 1 } , N _ { 2 }$ , and $N _ { 3 }$ Suppose $N _ { i } \cap N _ { j } \mathrm { ~ = ~ } \{ e \}$ and $N _ { i } N _ { j } = G$ for all $i , j$ with $i \neq j$ . Show that G is abelian and $N _ { i }$ is isomorphic to $N _ { j }$ for all $i , j$

Problem 13 Consider the system of differential equations:

$$
 { \frac { d x } { d t } } = y + t z
$$

$$
{ \frac { d y } { d t } } = z + t ^ { 2 } x
$$

$$
\frac { d z } { d t } = x + e ^ { t } y .
$$

Prove there exists a solution defined for all $t \in [ 0 , 1 ]$ , such that

$$
{ \begin{array} { r l } { { \left( \right)} { \begin{array} { l l l } { 1 } & { 2 } & { 3 } \\ { 4 } & { 5 } & { 6 } \\ { 7 } & { 8 } & { 9 } \end{array} }  } & { { \Bigg ( } { \begin{array} { l } { x ( 0 ) } \\ { y ( 0 ) } \\ { z ( 0 ) } \end{array} } { \Bigg ) } = { \binom { 0 } { 0 } } } \end{array} }
$$

and also

$$
\int _ { 0 } ^ { 1 } \left( x ( t ) ^ { 2 } + y ( t ) ^ { 2 } + z ( t ) ^ { 2 } \right) d t = 1 .
$$

Problem 14 Let $M _ { n \times n }$ denote the vector space of $n \times n$ real matrices for $n \geqslant 2$ . Let det : $M _ { n \times n } \to \mathbb { R }$ be the determinant map.

1. Show that det is $C ^ { \infty }$

2. Show that the derivative of det at $A \in M _ { n \times n }$ is zero $i f$ and only if A has rank $\leqslant n - 2$

Problem 15 Which of the following matrices are similar as matrices over R?

$$
( a ) ~ { \left( \begin{array} { l l l } { 1 } & { 0 } & { 0 } \\ { 0 } & { 1 } & { 0 } \\ { 0 } & { 0 } & { 1 } \end{array} \right) } ~ , ~ ( b ) ~ { \left( \begin{array} { l l l } { 0 } & { 0 } & { 1 } \\ { 0 } & { 1 } & { 0 } \\ { 1 } & { 0 } & { 0 } \end{array} \right) } ~ , ~ ( c ) ~ { \left( \begin{array} { l l l } { 1 } & { 0 } & { 0 } \\ { 1 } & { 1 } & { 0 } \\ { 0 } & { 0 } & { 1 } \end{array} \right) } ~ ,
$$

$$
( d ) \left( { \begin{array} { c c c } { 1 } & { 0 } & { 0 } \\ { 1 } & { 1 } & { 0 } \\ { 0 } & { 1 } & { 1 } \end{array} } \right) , \left( e \right) \left( { \begin{array} { c c c } { 1 } & { 1 } & { 0 } \\ { 0 } & { 1 } & { 0 } \\ { 0 } & { 0 } & { 1 } \end{array} } \right) , \left( f \right) \left( { \begin{array} { c c c } { 0 } & { 1 } & { 1 } \\ { 0 } & { 1 } & { 0 } \\ { 1 } & { 0 } & { 0 } \end{array} } \right) .
$$

Problem 16 For which $z \in \mathbb { C }$ does

$$
\sum _ { n = 0 } ^ { \infty } \left( { \frac { z ^ { n } } { n ! } } + { \frac { n ^ { 2 } } { z ^ { n } } } \right)
$$

converge?

Problem 17 Let P and Q be complex polynomials with the degree of Q at least two more than the degree of P . Prove there is an $r > 0$ such that if C is a closed curve outside $| z | = r$ , then

$$
\int _ { C } { \frac { P ( z ) } { Q ( z ) } } d z = 0 .
$$

Problem 18 Show that for any continuous function $f : [ 0 , 1 ] \to \mathbb { R }$ and $\varepsilon > 0$ , there is a function of the form

$$
g ( x ) = \sum _ { k = 0 } ^ { n } C _ { k } x ^ { 4 k }
$$

for some $n \in \mathbb { Z }$ , where $C _ { 0 } , \dots , C _ { n } \in \mathbb { Q }$ and $| g ( x ) - f ( x ) | < \varepsilon$ for all x in [0, 1].

Problem 19 Let P be $a \ n \times n$ real matrix such that xt $P y = - y ^ { t } P x$ for all column vectors x, y in $\mathbb { R } ^ { n }$ . Prove that P is skew-symmetric.

Problem 20 Give an example of a function $f : \mathbb { R } \to \mathbb { R }$ having all three of the following properties:

$f ( x ) = 0$ for $x < 0$ and $x > 2$

• $f ^ { \prime } ( 1 ) = 1$

• f has derivatives of all orders.