\documentclass{article}
\newtheorem{thm}{Theorem}
\newcommand{\inter}{{\rm int\,}}
\newcommand{\cl}{{\rm cl\,}}
\begin{document}
\begin{center}

{\large\bf C\&O 367 \\ 
Assignment 0 (REVISION) 
}
\end{center}
\begin{flushleft}
{\large  Due on Tuesday, Jan. 14, 2001.   Instructor H. Wolkowicz
}
\end{flushleft}
\bigskip

\begin{center}
\section{GEOMETRY}
\end{center}
\subsection{BACKGROUND MATERIAL}

For a vector $x=(x_1, \cdots, x_n)^t \in \Re^n$, we define the {\em
norm} and {\em inner product}
\[  \|x\| := (x_1^2+ \cdots+ x_n^2)^{1/2},
\]
\[  x \cdot y := x_1y_1+ \cdots+ x_ny_n.
\]
\begin{thm}
(Cauchy-Schwarz) $|x \cdot y| \leq \|x\|\|y\|,$ with equality if and
only if $x=\lambda y,$ for some $\lambda.$
\end{thm}
\subsection{EXERCISES}
\begin{enumerate}
\item (10)
Explain the geometrical significance for the vectors $x$ and $y$ of:
\begin{enumerate}
\item
\[x \cdot y = 0. \]
\item
\[x \cdot y \leq 0. \]
\item
\[ \cos \theta = \frac {x \cdot y}{\|x\|\|y\|},
\]
where $\theta$ is an angle.
\end{enumerate}
\item
(10) Prove that the function $f: \Re^n \rightarrow \Re$ defined by $f(x)
= a \cdot x+\alpha$ is continuous, where $a$ is a given vector and $\alpha \in
\Re$.
\item
(10) Prove that the function $f: \Re^n \rightarrow \Re$ defined by $f(x)
= \|x\|$ is continuous.
\end{enumerate}
\begin{center}
\section{CALCULUS}
\end{center}
\subsection{BACKGROUND MATERIAL}
For a function $g: \Re^n \rightarrow \Re$, the {\em gradient} is denoted
\[
\nabla g := \left( \begin{array}{c}
   \frac{\partial g}{\partial x_1}\\ 
   \frac{\partial g}{\partial x_2}\\ 
     \cdots\\ \frac{\partial
g}{\partial x_n}
\end{array}  \right).
\]

Suppose $f:\Re \rightarrow \Re$ is infinitely differentiable at $x=a.$
The {\em Taylor Series} of $f$ about $a$ is:
\[
f(a) + f^\prime(a) (x-a) + \frac1{2!}f^{\prime \prime}(a)(x-a)^2 + 
    \frac1{3!}f^{\prime \prime \prime}(a)(x-a)^3 + \cdots.
\]
\subsection{EXERCISES}
\begin{enumerate}
\item 
(5)
Let $g(x) = \| 7x+e \|$, where $x,e \in \Re^n$ and $e$ is the
vector of ones. Calculate the gradient of $g$, $\nabla g(x)$, when
$7x+e \neq 0$.
\item (10)
Suppose that $g:\Re^n \rightarrow \Re$, $a,b \in \Re^n$, and $f(t) := g(a-tb)$.
Calculate the derivative $f^\prime (t).$

\item
Write down the Taylor series of:
\begin{enumerate}
\item (5)
\[
f(x) = x^5+x^3, \mbox{~about~} x=1.
\]
\item (5)
\[
f(x) = \log(1+x), \mbox{~about~} x=0.
\]
\item (5)
\[
f(x) = x\cos(x), \mbox{~about~} x=0.
\]
\end{enumerate}

\end{enumerate}
\begin{center}
\section{TOPOLOGY}
\end{center}
\subsection{BACKGROUND MATERIAL}
The {\em open ball} $B(x;r) : = \left\{ y \in \Re^n : \|x-y\| < r
\right\}.$ Suppose that $D$ is a subset of $\Re^n$.
\begin{description}
\item[Interior:]
$x \in \inter D$ if there exists $r > 0$ with $B(x;r) \subset D$.
\item[Closure:]
$x \in \cl D$ if there exists a sequence $x^k \in D$ with $x^k
\rightarrow x.$
\item[Boundary:]
$x \in \partial D$ if $x\in \cl D \backslash \inter D.$
\end{description}
$D$ is {\em open} if $D = \inter D.$ 
$D$ is {\em closed} if $D = \cl D.$ \\
~~\\
\subsection{EXERCISES}
\begin{enumerate}
\item
(15)  For each of the following sets, find the interior, the
closure, and the boundary. Then determine which of the sets are open,
closed, neither, or both.
\begin{enumerate}
\item
\[
\left\{
(x_1,x_2) : x_1 \geq 0, x_2 \geq 0
\right\}.
\]
\item
\[
\left\{
(x_1,x_2) : x_1 > 0, x_2 > 0
\right\}.
\]
\item
\[
\left\{
(x_1,x_2) : x_1 > 0, x_2 \geq 0
\right\}.
\]
\item  
\[
\Re^n
\]
\item
\[
\left\{
(x_1,x_2) : x_1^2 + x_2^2 < 0
\right\}.
\]
\item
\[
\emptyset .
\]
\end{enumerate}
\item
\begin{enumerate}
\item (10) Prove that $D$ is closed if and only if the complement $D^c$
is open.
\item (10)
Prove that $x \in \partial D$ if and only if for any $r>0$ there exists
a $y \in B(x;r) \cap D$ and a $z \in B(x;r) \cap D^c.$
\end{enumerate}

\end{enumerate}

\begin{center}
\section{MATRICES}
\end{center}
\subsection{EXERCISES}
\begin{enumerate}
\item (10)
Let
\[ 
A = \left(
\begin{array}{ccc}
-1 & 2 & 3 \\
0 & 2 & 1 \\
1 & -2 & 2 
 \end{array}
\right)
\]

\begin{enumerate}
\item
Calculate the determinant of $A$.
\item
Calculate the trace of $A$.
\item
Calculate the rank of $A$.
\item
What is the rank of $A^T$.
\end{enumerate}
  

\item (10)
Let
\[ 
B = \left(
\begin{array}{cc}
5 & -3  \\
-3 & 2
 \end{array}
\right)
\]
Calculate the eigenvalues and eigenvectors of $B$.

\end{enumerate}
\end{document}
