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\hskip36pc \= Math.\ 362-01\\
\>Summer, 2005\\
\>R. B. Kearfott
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{\bf Second Exam} \\
{\em Thursday, June 30, 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 entire problem is worth 25 points.} \LARGE
\begin{enumerate}
   \item \label{prob:row-space}
         Write down a basis for the row space of the matrix
         $$A = \left[\begin{array}{rrrr}
               1 &  1 &  1 &  1 \\
              -1 &  1 & -1 &  1 \\
               1 &  2 &  3 &  1 \\
               2 &  5 &  4 &  4
               \end{array}\right].$$
   \item Is the null-space of the matrix $A$ above non-trivial?
         (That is, does the null space contain any non-zero
         vectors?)  If so, then write down a basis for the null
         space of this matrix.
   \item In each of the following, justify the answers that you
         give.
         \begin{enumerate}
            \item What is the rank of the matrix $A$?
            \item What is the dimension of the null space of the
                  matrix $A$?
            \item What is the dimension of the column space of the
                  matrix $A$?
         \end{enumerate}
   \item Determine whether or not the vector
         $\left(\begin{array}{rrrr}3 & 2 & 5 & 1
         \end{array}\right)^t$
         lies in the row space of the matrix
         $A$.  Show all of your
         computations and / or write down your reasoning
         completely.
\end{enumerate}

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