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\begin{center}

{\large\bf C\&O 367 \\ 
Assignment 0 (REVISION) 
}
\end{center}
\begin{flushleft}
{\large  Due on Thursday, Jan. 14, 1999.   Instructor H. Wolkowicz
}
\end{flushleft}
\bigskip

\begin{enumerate}
\item
\begin{center}
{\large\bf GEOMETRY
}
\end{center}
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}
{\bf EXERCISES}
\begin{enumerate}
\item (10)
Explain the geometrical significance for the vectors $x$ and $y$ of:
\begin{enumerate}
\item
\[x \cdot y = 0. \]
\item
\[x \cdot y > 0. \]
\end{enumerate}
\item
(10) Prove that the function $f: \Re^n \rightarrow \Re$ defined by $f(x)
= a \cdot x$ is continuous, where $a$ is a given vector.
\end{enumerate}
\item
\begin{center}

{\large\bf CALCULUS
}
\end{center}
For a function $g: \Re^n \rightarrow \Re$, the {\em gradient} is denoted
\[
\nabla g := \left( \begin{array}{c}
   \frac{\partial g}{\partial x_1}\\ 
   \frac{\partial g}{\partial x_2}\\ 
     \cdots\\ \frac{\partial
g}{\partial x_n}
\end{array}  \right)
\]
{\bf EXERCISES}
\begin{enumerate}
\item 
(5)
If $g(x) = || x ||,$ calculate $\nabla g(x).$
\item (10)
Suppose $f:\Re^n \rightarrow \Re,~ a,b \in \Re^n,$ and $f(t) := g(a+tb)$.
Calculate $f^\prime (t).$
\end{enumerate}
Suppose $f:\Re \rightarrow \Re$ is infinitely differentiable at $x=a.$
The {\em Taylor Series} of $f$ about $a$ is:
\[
f(a) + f^\prime(a) (x-a) + \frac1{2!}f^{\prime \prime}(a)(x-a)^2 + 
    \frac1{3!}f^{\prime \prime \prime}(a)(x-a)^3 + \cdots
\]
{\bf EXERCISE}\\
Write down the Taylor series of:
\begin{enumerate}
\item (5)
\[
f(x) = x^3, \mbox{~about~} x=1.
\]
\item (5)
\[
f(x) = \log(1+x), \mbox{~about~} x=0.
\]
\end{enumerate}
\item
\begin{center}

{\large\bf TOPOLOGY
}
\end{center}
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 $D = \cl D.$ \\
~~\\
{\bf 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}
\item
\begin{center}
{\large\bf MATRICES
}
\end{center}
{\bf EXERCISES}\\
\begin{enumerate}
\item (10)
Let
\[ 
A = \left(
\begin{array}{ccc}
1 & 1 & 0 \\
0 & 1 & 1 \\
1 & 2 & 1 
 \end{array}
\right)
\]

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

\item (10)
Let
\[ 
B = \left(
\begin{array}{cc}
2 & 1  \\
1 & 2
 \end{array}
\right)
\]
Calculate the eigenvalues and eigenvectors of $B$.

\end{enumerate}
\end{enumerate}
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