\be
\item Suppose that $A$ is an $m$-by-$n$ matrix whose rows form a linearly independent
set of vectors. Show that the $m$-by-$m$ matrix $AA^t$ is invertible. (Hint: Start with
the homogeneous matrix equation $AA^t \vecx=\zerovec$. Multiply each side by $\vecx^t$, the transpose
vector of $\vecx$. Use the result to establish $\left\| A^t \vecx \right\|=0$, and then apply the Invertible Matrix Theorem.)
\item Show that the projection matrix, $P$, and projected gradient, $\vecc_p$,
as given in Definition \ref{Th:ProjectedGradient} satisfy the following three properties:
\be
\item The vector $\vecc_p$ belongs to the null space of $A$. That is, $A\vecc_p=\zerovec$.
\item The matrix $P$ is symmetric, meaning that $P^t=P$.
\item The matrix $P$ satisfies $P^2\vecc^t=P\vecc^t$.
\ee

\item Assume in the \LP LP that $\ds \vecx_0= \begin{bmatrix} 3 \\ 4 \\ 5 \\ 14 \\15 \end{bmatrix}$, which corresponds to $\begin{bmatrix}  3 \\ 4 \end{bmatrix}$ in $x_1x_2$-plane. If we denote
    $\tilde{\vecx_k} = \begin{bmatrix}  \tilde{x}_1 \\ \tilde{x}_2 \end{bmatrix}$, sketch the image in the $\tilde{x}_1\tilde{x_2}$-plane of each basic feasible solutions of the \LP LP under the first centering transformation $D^{-1}$. Show that the region in the $\tilde{x}_1\tilde{x_2}$-plane enclosed by these points coincides with the feasible region of the LP,
    \begin{align*}
		 \mbox{maximize } z &=\tilde{\vecc} \tilde{\vecx}  \\
		\mbox{ subject to} \notag\\
		\tilde{A}\tilde{\vecx} \leq \vecb \\
		\tilde{\vecx} &\geq \zerovec, \notag
		\end{align*}
    where $\tilde{A}$ and $\tilde{\vecc}$ are as defined in step 3 of Section \ref{S:InteriorMethod_Fuelpro}.

\item Approximate, to three decimal places, the solutions of the following LPs.
\be
\item
		\begin{align*}
		 \mbox{maximize } z &=3x_1+2x_2  \\
		\mbox{ subject to} \notag\\
		x_1 \phantom{ +20x_2} &\leq 4 \notag\\
		x_1+3x_2 &\leq 15  \notag\\
		2x_1+x_2 &\leq 10 \notag \\
		x_1,x_2 &\geq 0\notag
		\end{align*}	
	
	\item
		\begin{align*}
		 \mbox{maximize } z &=4x_1 + x_2 +5x_3  \\
		\mbox{ subject to} \\
		2x_1+x_2+3x_3 &\leq 14 \\
		6x_1+3x_2 +3x_3 &\leq 22  \\
		2x_1+3x_2  \x &\leq 14  \\
		x_1,x_2,x_3 &\geq 0
		\end{align*}	
\ee

\item Use the interior point algorithm to approximate the solution of the
\emph{Foraging Herbivore Model}\index{Foraging Herbivore Model}, from Exercise 4 of Section
\ref{S:LP_Prototype_Standard_Form}.

\ee


