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\title{General Mathematics and Computational Science II}
\author{Exercise 14}
\date{April 12, 2007}
\maketitle

\begin{enumerate}

\item Find all solutions for the underdetermined linear system
$A\boldsymbol{x}=\boldsymbol{b}$, 
where
\begin{equation*}
  A =
  \begin{pmatrix}
    2 & 2 & 1 & 0 \\
    1 & 1 & 1 & 1 \\
    1 & 1 & 0 & -1 \\
    3 & 3 & 2 & 1
  \end{pmatrix}
  \qquad \text{and} \qquad
  \boldsymbol{b} =
  \begin{pmatrix}
    -1 \\ 1 \\ -2 \\ 0
  \end{pmatrix} \,.
\end{equation*}

\item Write the following linear programming problem in its standard
form.  

Maximize
\[
  z = 2 \, x_1 - x_2 + x_3
\]
subject to 
\begin{gather*}
  x_1 - x_2 \leq 1 \,, \\
  x_2 - x_3 \geq 1 \,, \\
  x_3 - x_1 \leq 3 \,, \\
  x_1 \geq -2 \,, \\
  x_2 \leq 1 \,, \\
  x_3 \leq 0 \,.
\end{gather*}

\item Assume that $\boldsymbol{x}, \boldsymbol{y} \in \mathbb{R}^n$
are two solutions to the linear programming problem in standard form,
\begin{gather*}
  \text{minimize } z = \boldsymbol{c}^T \boldsymbol{x} \\
  \text{subject to } A \boldsymbol{x} = \boldsymbol{b} \\
  \text{and } \boldsymbol{x} \geq \boldsymbol{0} \,.
\end{gather*}

Show that any convex combination $\boldsymbol{z} = t \, \boldsymbol{x}
+ (1-t) \, \boldsymbol{y}$ for $ t \in [0,1]$ is also a solution.

\end{enumerate}
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