# 2017 GRADUATE PRELIMINARY EXAM

All problems are weighted equally. Throughout R denotes the real numbers and C the complex numbers.

(1) Negate the following statements in a "non-cheap" way  especially, avoid using the word "not."

(a) For all real numbers x, there is a real number y such that $| x - y | \geq 2 0 1 7 .$

The function $\pmb { f } : \mathbb { R }  \mathbb { R }$ is continuous.

) Let $V = \{ ( x , y , z ) \in \mathbb { R } ^ { 3 } \mid 3 x + 4 y + 5 z = 0 \}$

(a) Show that V is a linear subspace of $\mathbb { R } ^ { 3 }$

(b) Prove or disprove: there is a linear transformation $\pmb { S } : \mathbb { R } ^ { 3 }  \mathbb { R } ^ { 3 }$ with kernel equal to V.

Prove or disprove: there is a linear transformation $T : \mathbb { R } ^ { 3 }  \mathbb { R } ^ { 3 }$ with image equal to V.

(d) Prove or disprove: there is a linear transformation $\boldsymbol { U } : \mathbb { R } ^ { 3 }  \mathbb { R } ^ { 3 }$ with kernel and image equal to V.

(3) Show (we suggest by induction) that for all non-negative integers n, we have

$$
\int _ { 0 } ^ { \infty } x ^ { n } e ^ { - x } d x = n !
$$

(4) For a positive integer n, let $I _ { n }$ denote the $\pmb { n } \times \pmb { n }$ identity matrix.

() Let A be $\tt { a } 2 \times 2$ real matrix with eigenvalues $\lambda _ { 1 } = 1$ and $\lambda _ { 2 } = - 1$ .Show: $A ^ { 2 } = I _ { 2 }$

Find ${ \mathfrak { a } } 3 \times { \mathfrak { a } }$ real matrix whose only eigenvalues (in C) are 1 and -1 such that $A ^ { 2 } \neq I _ { 3 }$

(c) Let A be an n x n real symmetric matrix whose only eigenvalues (in C) are 1 and -1. Show that $A ^ { 2 } = I _ { n }$ •

5) Let $\{ f _ { n } : [ 0 , 1 ] \to \mathbb { R } \} _ { n = 1 } ^ { \infty }$ be a sequence of continuous functions that converges uniformly to 0. Show that the sequence $\begin{array} { r } { \int _ { 0 } ^ { 1 } f _ { n } ( x ) d x } \end{array}$ converges to 0.

Use the $\epsilon , \delta$ definition of the limit to show: $\begin{array} { r } { \operatorname* { l i m } _ { x \to 1 } \frac { x ^ { 2 } + 1 } { x } = 2 . } \end{array}$

(7) Let ${ z } = f ( x , y )$ be a smooth surface. Show that the gradient is perpendicular to the level curves. (Suggestion: let $\gamma ( t )$ be a curve contained in a level set of $f ,$ and consider the derivative of $f \circ \gamma . )$

(8) Let X and Y be sets and let $f : X \to Y$ and $g : Y \to X$ be functions. We suppose throughout tat $g ( f ( x ) ) = x$ for all $\pmb { x } \in \pmb { X }$

(a) Show that f is injective.

Show that g is surjective.

(c) Give an example in which neither f nor g is bijective.