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\begin{document}
\title{Math 241 Homework}
\author{Mary Radcliffe}
\date{Due 13 September 2018}
\maketitle

Complete the following problems. Fully justify each response.

\begin{enumerate}

\item Complete problems 3.8.8, 3.8.9 on page 147 of Coding the Matrix.

\item Complete problems 4.17.2, 4.17.3 on page 207 of Coding the Matrix.

\item For each of the following matrices, describe how $\vec{v}$ and $A\vec{v}$ are related for an arbitrary vector $\vec{v}\in\R^n$.
\begin{enumerate}
\item $A = \left[\begin{array}{ccc}0 & 0 & 1\\ 0 & 1 & 0\\ 1 & 0 & 0\end{array}\right]$, $n=3$
\item $A=\left[\begin{array}{cccccc}\frac{1}{2}& \frac{1}{2} & 0 & 0 & 0 & 0\\ \frac{1}{2} & \frac{1}{2}& 0&0&0&0\\ 0&0&\frac{1}{2}&\frac{1}{2}&0&0\\0&0&\frac{1}{2}&\frac{1}{2}&0&0\\0&0&0&0&\frac{1}{2}&\frac{1}{2}\\0&0&0&0&\frac{1}{2}&\frac{1}{2}\end{array}\right]$, $n=6$
\item $A=\left[\begin{array}{ccc}1 & 1 & 1\\ 1&1&1\\1&1&1\end{array}\right]$, $n=3$
\end{enumerate}

\item Prove that if $M$ is an $m\times n$ matrix, and $\vec{u}, \vec{v}$ are both vectors of length $n$, that $M(\vec{u}+\vec{v}) = M\vec{u}+M\vec{v}$.


\item Complete the second problem set found at \url{autolab.andrew.cmu.edu}. The submission for this is directly on autolab, no need to hand it in on paper.
\end{enumerate}
\end{document}