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\begin{tabbing}
\hskip36pc \= Math.\ 362-01\\
\>Summer, 2005\\
\>R. B. Kearfott
\end{tabbing}
\begin{center}
{\bf Third Exam} \\
{\em Thursday, July 14, 2005}
\end{center}

\noindent{\small This exam is closed book. Make sure your name is
on all pages.  You should put your work on your own paper, and you
may keep this exam sheet upon leaving.  Be sure to check your work
carefully, and to show intermediate computations in a logical
presentation.  \textbf{Full credit will not be given unless
computations are shown.} Each entire problem is worth 25 points.}

\begin{figure}[h]
\centerline{\mbox{ \includegraphics[scale=0.6]{tst3fig1.eps} }}
\caption{\label{fig:tst3fig1}The image for
problem~\protect{\ref{prob:map-image}}}
\end{figure}

\begin{enumerate}
   \item \label{prob:map-image}
         Write down the matrix for a linear transformation that maps
         the circle
         $x^2+y^2=1$ onto the ellipse in Figure~\ref{fig:tst3fig1}.
         Show all of your work.
   \item \label{prob:map-of-square}
         Draw the image of the unit square $0\le x_1\le 1$,
         $0\le x_2\le 1$ under the transformation $x\leftarrow Ax$,
         where
         $$ A = \left(\begin{array}{rr} 2 & -3 \\  2 & 3\end{array}
            \right).$$
         Label the sizes of the parts of the image and/or the coordinates
         of the vertices of the image.
   \item For the matrix $A$ as in Problem~\ref{prob:map-of-square},
         do the following:
         \begin{enumerate}
            \item Compute $A^{-1}$.
            \item Use $A^{-1}$ to find the solution $x$ to
                  $A x = (2,2)^{\mbox{t}}$.
            \item Use $A^{-1}$ to find the solution $x$ to
                  $A x = (-3,3)^{\mbox{t}}$.
         \end{enumerate}
   \item For the matrix $A$ from Problem~\ref{prob:map-of-square},
         do the following:
         \begin{enumerate}
            \item Compute the LU factorization of $A$.
            \item Use the LU factorization to find the solution $x$ to
                  $A x = (2,2)^{\mbox{t}}$.
            \item Use LU factorization to find the solution $x$ to
                  $A x = (-3,3)^{\mbox{t}}$.
         \end{enumerate}
\end{enumerate}

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