\begin{center}
Matrix Theory \\
Midterm Examination II\\[.2in]
\end{center}
Fall 1990 \\[.2in]
{\em Instructions:}  There are five (5) questions on this
examination, worth a total of 100 points. Work all problems.  \\[.1in]
\begin{enumerate}

	\item  (20 points)\\
Solve the system:\\ 
     \begin{tabular}{rrrrrrrrr}
\qquad \qquad   $c_0$ & + & $c_1$ & + & $c_2$ & + & $c_3$ & = & 0   \\
\qquad \qquad   $c_0$ & + & i\ $c_1$ & - & $c_2$ & - & i\ $c_3$ & = & 1 \\
\qquad \qquad   $c_0$ & - & $c_1$ & + & $c_2$ & - & $c_3$ & = & 2  \\   
\qquad \qquad   $c_0$ & - & i\ $c_1$ & - & $c_2$ & + & i$c_3$ & = & 4 .   
\end{tabular}
Give the inverse of the coefficient matrix $W$.
\newpage          

	\item 
\begin{enumerate}
\item (10 pts)
If $P$ is the matrix that projects ${\cal{R}}^n$ onto a subspace $S$,
explain why every vector in $S$ is an eigenvector and so is every vector 
in $S^{\perp}$. What are the eigenvalues?\\[3.0in] 
\item (10 pts)
Find the eigenvalues and eigenvectors and the diagonalizing matrix $S$
for \\
\begin{eqnarray*}
A = \left[\begin{array}{rr}
          1 & 0\\
          2 & 3 \end{array}\right] & and & 
B = \left[\begin{array}{rr}
          7 & 2\\
          -15 & -4 \end{array}\right] 
\end{eqnarray*}

\end{enumerate}
\newpage


	\item  (20 points) \\
The following system has no solution:\\
\begin{eqnarray*}
Ax  =  \left[ \begin{array}{rr}
         1  &  -1 \\
         1  & 0 \\ 
         1  & 1 
           \end{array} \right]\ 
\left[ \begin{array}{r}
         C  \\
         D   
           \end{array} \right] & = &
\left[ \begin{array}{r}
         4   \\
         5   \\
         9
           \end{array} \right]  =  b.
\end{eqnarray*}
Sketch and solve a straight line fit that leads to the minimization of the 
quadratic $(C-D-4)^2 + (C-5)^2 + (C+D-9)^2$. What is the projection of $b$
onto the column space of $A$?
\newpage
\item
\begin{enumerate}
	\item (7 points)
Find an orthonormal basis for the column space of 
\[A = \left[ \begin{array}{rrrrr}
        1 & 3 & 4 & 5 & 7 \\
        -6 & 6 & 8 & 0 & 8
             \end{array}\right]^{T}. \]
        \item (7 points)
Write $A$ as $QR$, where $Q$ has orthonormal columns and $R$ is upper
triangular. 
        \item (6 points)
Find the least squares solution to $Ax = b$, if $b = (-3, 7, 1, 0, 4)^T$.

\end{enumerate}
\newpage
\item
\begin{enumerate}

	\item  (10 points)
If $A$ is nonsingular, show that there must be some permutation $P$
for which $PA$ has no zeroes on its main diagonal.
\item (10 points)
Explain why the point $(x,y)$ is on the line through $(2,8)$
and $(4,7)$ if
\begin{eqnarray*}
\mbox{det}\left[\begin{array}{rrr}
                       x & y & 1 \\
                       2 & 8 & 1 \\
                       4 & 7 & 1 \end{array}\right] = 0, & 
\mbox{or} & x + 2y = 18.
\end{eqnarray*}
\end{enumerate}
\end{enumerate}
\newpage
