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\hskip36pc \= Math.\ 362-01\\
\>Summer, 2005\\
\>R. B. Kearfott
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{\bf First Exam} \\
{\em Thursday, June 16, 2005}
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\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.  Each question is worth 15 points, and 10 points are
free. }

\Large Consider the system of equations
\begin{equation} \label{eq:first-system}
\begin{array}{rcrcrcrcr}
 x_1  &+& x_2 &+&  x_3 &+&  x_4 &=& 3 \\
 x_1  &-& x_2 &+&  x_3 &-&  x_4 &=& 3 \\
2x_1  & &     &+& 2x_3 & &      &=& 6
\end{array}
\end{equation}

\begin{enumerate}
   \item Write down the augmented matrix for the system of equations
         \eqref{eq:first-system}.
   \item Put this matrix into reduced row-echelon form.
   \item What is the rank of this matrix?
   \item Write down the solution of the system in terms of spanning
         vectors and a translation vector (that is, in parametric
         form).
   \item Does the solution represent a point, line, plane, or some
         higher-dimensional object?
   \item Write down a matrix $A$ and a vector $b$ such that $A x =
         b$ corresponds to the system \eqref{eq:first-system},
         where $x = [x_1, x_2, x_3, x_4]^T$.

\end{enumerate}

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