\begin{center}
Matrix Theory \\
Examination I\\[.2in]
\end{center}
Fall 1997 \\[.2in]
{\em Instructions:}  There are five (5) questions on this
examination, worth a total of 100 points. Work all problems.  \\[.1in]
\begin{enumerate}

	\item  (15 points)\\
 (i) Solve by elimination and back substitution:\\ 
     \begin{tabular}{ccccccccccccccc}
\qquad \qquad   u & ~ & ~ & + & w & = & 4 & \qquad and 
       \qquad & ~ & ~ & v & + & w & = & 0   \\
\qquad \qquad   u & + & v & ~ & ~ & = & 3 & \qquad and 
       \qquad & u & ~ & ~ & + & w & = & 0   \\
\qquad \qquad   u & + & v & + & w & = & 6 & \qquad and 
       \qquad & u & + & v & ~ & ~ & = & 6   
\end{tabular}
 \\[.05in]
 (ii) Factor the above matrices into $A = LU$ or $PA = LU$.
\newpage          

	\item (15 points)  \\
(i) If $A$ is square, show that the nullspace of $A^2$ contains the
nullspace of $A$.\\
(ii) Show also that the column space of $A^2$ is contained in the
column space of A.\\
(iii)If $e_1$,$e_2$,$e_3$ are in the column space of a $3 \times 5$
matrix, does it have a right inverse? 
\newpage


	\item  (30 points)
 Suppose the matrices in $PA = LU$ are:
\begin{eqnarray*}
\left[ \begin{array}{cccc}
       0 & 1 & 0 & 0 \\
       1 & 0 & 0 & 0 \\
       0 & 0 & 0 & 1 \\
       0 & 0 & 1 & 0 
       \end{array}  \right] 
\left[ \begin{array}{crcrc}
       0 &  0 & 1 & -3 & 2 \\
       2 & -1 & 4 &  2 & 1 \\
       4 & -2 & 9 &  1 & 4 \\
       2 & -1 & 5 & -1 & 5 
       \end{array}  \right] \\
= \left[ \begin{array}{cccc}
        1 & 0 & 0 & 0 \\
        0 & 1 & 0 & 0 \\
        1 & 1 & 1 & 0 \\
        2 & 1 & 0 & 1 
         \end{array}  \right] 
  \left[ \begin{array}{crcrc}
        2 & -1 & 4 &  2 & 1 \\
        0 &  0 & 1 & -3 & 2 \\
        0 &  0 & 0 &  0 & 2 \\
        0 &  0 & 0 &  0 & 0 
         \end{array}  \right] .
\end{eqnarray*}
(i) What is the rank of $A$?\\
(ii) What is a basis for ${cal{R}}(A^T)$?\\
(iii) TRUE or FALSE: Rows 1, 2, 3 of $A$ are linearly independent.\\
(iv) What is a basis for ${\cal{R}}(A)$?\\
(v) What is the dimension of ${\cal{N}}(A^T)$?\\
(vi) What is the general solution to $Ax = 0$?

\newpage

        \item  (20 points)\\
\begin{enumerate}
        \item
 Show that if the $n\times n$ matrix $A$ is symmetric with positive pivots, 
then there exists a lower triangular $n\times n$ matrix $L$  with positive
diagonal entries such that  $ B = L L^T$ (Cholesky factorization).
        \item 
Find the Cholesky factorization for the matrix
\[ B = \left[ \begin{array}{rr}
                   4 & 12 \\
                  12 & 45    \end{array} \right] .\]
        \item
Employ the Cholesky factorization to 
find a least squares solution to the inconsistent system
\[  \left[ \begin{array}{rr}
           1 & 2 \\ 3 & 4 \\ 5 & 6 \end{array} \right] x =
    \left[ \begin{array}{rr}
           1 \\ 1 \\ 1 \end{array} \right]  . \]
(Hint: the matrix $A^TA$ is symmetric.)
\end{enumerate}

\newpage 

\item   (20 points)\\
Consider the LS problem
\[\left( \begin{array}{rr} 
        1 & 2 \\
        0 & 1 \\
        1 &-1 \\
       -1 & 3 \end{array} \right) x = 
  \left( \begin{array}{r} 
         0 \\ 1 \\ 0 \\ 0 \end{array} \right) . \]
Find the Least squares solution by using the projection method.

\end{enumerate}
