# Graduate preliminary examination, Fall 2003 3 hours, 8 problems counted equally

#1. a) Let f be a function defined on an interval $( a , b )$ and let $c \in \left( a , b \right)$ . Define what it means to say that $f$ is continuous at c.

b) Use this definition to show that $f ( x ) = 1 / x$ is continuous at $x = 1$

#2. Find a $2 \times 2$ matrix P such that $P \left( { \begin{array} { l l } { 1 } & { 2 } \\ { { \frac { 2 } { 9 } } } & { 1 } \end{array} } \right) P ^ { - 1 }$ is diagonal.

#3. Show that the function

$$
f ( x ) = { \left\{ \begin{array} { l l } { x ^ { 2 } \sin { \frac { 1 } { x } } , { \mathrm { ~ f o r ~ } } x \neq 0 } \\ { 0 , { \mathrm { ~ f o r ~ } } x = 0 } \end{array} \right. }
$$

is differentiable everywhere in $( - \infty , \infty )$ , but that $f ^ { \prime }$ is not continuous at 0.

#4. Let M be an $m \times n$ matrix, and let $V = \{ v \in \mathbb { R } ^ { n } : M v = 0 \}$ and $W = \{ M ^ { t } y : y \in \mathbb { R } ^ { m } \}$

a) Prove that V and W are vector subspaces of $R ^ { n }$

Prove that $V = \{ x \in \mathbb { R } ^ { n } : x \cdot w = 0$ for all $w \in W \}$

#5. Prove by induction that the sum of the cubes of 3 consecutive positive integers is divisible by 9.

#6. a) Define what is meant for an infinite series $\sum _ { n = 1 } ^ { \infty } b _ { n }$ of real numbers bn to converge.
$b _ { n }$ Let $a _ { 1 } , a _ { 2 } , \dotsc$ be a sequence of positive real numbers such that

$$
a _ { 1 } > a _ { 2 } > \cdots > a _ { n } > a _ { n + 1 } > \ldots \mathrm { ~ a n d ~ } \operatorname* { l i m } _ { n \to \infty } a _ { n } = 0 .
$$

Prove that the infinite series $\scriptstyle \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } a _ { n }$ converges.

#7. a)Let $\gamma$ be a path in the plane $\mathbb { R } ^ { 2 }$ . Define what is meant by

$$
\int _ { \gamma } f ( x , y ) d x + g ( x , y ) d y .
$$

b) Compute this line integral in the case where $f ( x , y ) = 2 x y , g ( x , y ) = x ^ { 2 } + y ^ { 2 }$ and $\gamma$ is the straight-line path from $P = ( 1 , 0 )$ to $Q = ( 0 , 1 )$

c) Is there another path $\beta$ from P to Q such that the corresponding line integral takes a different value?

#8. Let A be an $n \times n$ matrix.

a) Define what is meant by the eigenvalues and eigenvectors of A.

b) Show that if v and w are eigenvectors of A corresponding to distinct eigenvalues λ and $\mu ,$ then v and w are linearly independent.