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\lhead{\large\sc APMA 308 } \chead{\large\sc Midterm \#1}
\rhead{\large\sc February 26\textsuperscript{th}, 2008} \cfoot{}

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{\sc Name:} \ \rule{8cm}{.01cm}  \\
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{\sc Pledge:}   \hrulefill \\[3mm]
\rule{0cm}{0cm}  \hrulefill \\[3mm]
\hspace*{2in}   {\sc Signature:} \ \hrulefill
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{\em To get credit for a problem, you must show all of your
reasoning and calculations.  No calculators may be used.}
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\hfill \begin{tabular}{c|c} {\bf Problem} &   {\bf Score}
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1 &
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2 &
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3 &
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4 &
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5 &
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{\bf Total} &
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\begin{enumerate}
\item  (20 pts) Solve the following homogeneous system of linear equations and
find a basis for and the dimension of the subspace of the solutions.
\begin{align*}
x_1+x_2-2x_3+3x_4=&0 \\
2x_1+x_2-5x_3+2x_4=&0 \\
3x_1+x_2-8x_3+x_4=&0
\end{align*}

\newpage


\item  $(20pts)$ Solve the system of linear equations by
determining the inverse of the matrix of coefficients and then using
matrix multiplication.
\begin{align*}
x_1+2x_2+x_3=&1 \\
3x_1+5x_2-3x_3=&0 \\
2x_1+3x_2-3x_3=&2
\end{align*}


\newpage

\newpage
\item  (20pts) Solve the following system of linear equations
using the method of $LU$ decomposition.
\begin{align*}
2x_1-x_2+x_3=&4 \\
4x_1-x_2+4x_3=&0 \\
2x_1+x_3=&2
\end{align*}



\newpage
\item  \begin{enumerate}
\item (12pts) Find a basis for the subspace of all vectors orthogonal to
both $(1,2,1,2)$ and $(2,5, 3, 1)$.
\item (8pts) Check whether the vectors $(1,1,1)$, $(1,1,-1)$, and
 $(-1,1,1)$ are linearly independent or not
\end{enumerate}

\newpage
\item \begin{enumerate}
\item (6pts) Construct a system of linear equations that describes
the traffic flow in the road network of the following figure. (need not solve).
\[
\xymatrix{
{} & {} & {} & {\bullet} \ar@(r,u)[rrd]^{x_4} \ar[d]^{x_5} & {}  & {} & {}\\
{}\ar[r]^{150} & {\bullet} \ar@(u,l)[urr]^{x_1} \ar[rr]^{x_2} \ar@(d,l)[drr]_{x_3} & {} & {\bullet}\ar[rr]^{x_8} & {} & {\bullet}\ar[r]^{150} & {} \\
{} & {} & {} & {\bullet} \ar@(r,d)[rru]_{x_6} \ar[u]_{x_7} & {} & {} & {}
}
\]

\item (6pts) Check whether $T: \Bbb {R}^2 \longrightarrow \Bbb {R}^2,
T(x,y) = (x+y, xy)$ is a linear transformation or not.
\item (8pts) Show that $W=\{(a,b,c) | a+b+c=0; a, b, c \in \Bbb{R}\}$ is
a subspace.
\end{enumerate}




\end{enumerate}
\end{document}
