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\begin{document}
\title{Math 241 Homework}
\author{Mary Radcliffe}
\date{Due 11 October 2018}
\maketitle

Complete the following problems. Fully justify each response.

\begin{enumerate}

\item Read sections 5.4.1, 5.4.3, and 5.5.3 in Coding the Matrix about Minimum Spanning Forest. Then complete exercises 5.14.4 and 5.14.10 on pages 254-255.

\item For each of the following sets of vectors over $\R$, determine if the set is linearly dependent or independent. If it is linearly dependent, find a maximum subset that is linearly independent.
\begin{enumerate}
\item $\left\{ \threevec{1}{2}{1}, \threevec{2}{0}{-1}, \threevec{0}{1}{1}, \threevec{3}{1}{-1}\right\}$
\item $\left\{ \fourvec{1}{0}{1}{0}, \fourvec{0}{1}{0}{1}, \fourvec{1}{0}{-1}{0}, \fourvec{0}{1}{0}{-1}\right\}$
\item $\left\{ \fourvec{1}{0}{-1}{1}, \fourvec{0}{1}{-1}{1}, \fourvec{1}{1}{0}{0} \right\}$
\end{enumerate}

\item Suppose that $\left\{\vec{a}, \vec{b}, \vec{c}\right\}$ is a set of linearly independent vectors. 
\begin{enumerate}
\item Is $\{\vec{a}+\vec{b}, \vec{b}+\vec{c}, \vec{a}+\vec{c}\}$ linearly independent? Either prove that it is, or give a counterexample to demonstrate that it is not.
\item Is $\{\vec{a}-\vec{b}, \vec{b}-\vec{c}, \vec{a}-\vec{c}\}$ linearly independent? Either prove that it is, or give a counterexample to demonstrate that it is not.
\end{enumerate}

\item Let $S$ be a set of vectors, and let $A\subseteq S$. Suppose $\vec{z}\in \sp{S}$, and that $A\cup\{\vec z\}$ is linearly independent. Prove that there is a vector $\vec{w}\in S\ba A$ such that $\sp{S} = \sp{S\ba\{\vec{w}\}\cup\{\vec{z}\}}$.

NOTE: Basically, what we are saying in this proposition is that we can exchange vector $\vec{z}$ for $\vec{w}$ in $S$, without changing the span. The part about set $A$ is essentially to protect a set of vectors in $S$ from being replaced by this process. So we have a set $S$, some vectors we want to keep (namely $A$), and a vector $\vec{z}$ we want to include. The lemma says that there's some $\vec{w}\in S\ba A$ such that we can switch out $\vec{w}$ for $\vec{z}$ under the condition that $A\cup \{\vec{z}\}$ is linearly independent. This condition, of course, is not necessary, but it is sufficient.

\item Write a change of basis matrix that will change coordinates from $\mathcal{B}$ to $\mathcal{C}$, two different bases for $\R^3$, listed below.
\[\mathcal{B} = \left\{ \threevec{-1}{1}{1}, \threevec{1}{-1}{1}, \threevec{1}{1}{-1} \right\}\]

\[\mathcal{C} = \left\{ \threevec{1}{1}{0}, \threevec{1}{0}{1}, \threevec{0}{1}{1} \right\}\]

\item Let $A$ be an $n\times n$ matrix over $\R$, and suppose $\vec{v}_1, \vec{v}_2, \dots, \vec{v}_k$ are nonzero vectors satisfy the following:
\begin{itemize}
\item For each $i$, $A\vec{v}_i=\lambda_i\vec{v}_i$ for some $\lambda_i\in \R$
\item If $i\neq j$, then $\lambda_i\neq \lambda_j$
\end{itemize}
Prove that $\{\vec{v}_1, \dots, \vec{v}_k\}$ is linearly independent. 

(Hint: try induction on $k$).

\item Complete the fifth 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}