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\headers{Math343}{Homework 4}{Due Sep 17, 2025}

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{%
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{%
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\noindent
\textbf{Directions:} You may work to solve these problems in groups, but all written work must be your own.  Show all work; no credit for solutions without work..

\begin{enumerate}

\item {}[1.4.14] Let $\vect{u} = \MTBO{2}{-3}{2}$ and $A = 
\left[
\begin{array}{*{3}{r}}
5 & 8 & 7\\
0 & 1 & -1\\
1 & 3 & 0
\end{array}
\right]$.  Is $\vect{u}$ in the subset of $\RR^3$ spanned by the columns of $A$?  Why or why not?

\item {}[1.4.\{15,16\}] For the matrices $A$ and vectors $\vect{b}$ below, show that the equation $A\vect{x} = \vect{b}$ does not have a solution for all possible $\vect{b}$, and describe the set of all $\vect{b}$ for which $A\vect{x}=\vect{b}$ does have a solution.
\begin{enumerate}
	\item $A=\kgmmatrix{2}{2 & -1\\ -6 & 3}$ and $\vect{b}=\kgmmatrix{1}{b_1 \\ b_2}$.
	\item $A=\kgmmatrix{3}{1 & -3 & -4\\ -3 & 2 & 6\\ 5 & -1 & -8}$ and $\vect{b}=\MTBO{b_1}{b_2}{b_3}$
\end{enumerate}

\item {}[1.4.42] Could a set of three vectors in $\RR^4$ span all of $\RR^4$?  Explain.  What about $n$ vectors in $\RR^m$ when $n$ is less than $m$?

\item {}[1.5.\{7,11\}] Given a matrix $A$ and a vector $\vect{b}$, describe all solutions to $A\vect{x} = \vect{b}$ in parametric form.
\begin{enumerate}
	\item $A = \kgmmatrix{4}{1 & 3 & -3 & 7\\ 0 & 1 & -4 & 5}$, $\vect{b}=\vect{0}$
	\item $A = %
	\kgmmatrix{6}{1 & -4 & -2 & 0 & 3 & -5\\
	0 & 0 & 1 & 0 & 0 & -1\\
	0 & 0 & 0 & 0 & 1 & -4\\
	0 & 0 & 0 & 0 & 0 & 0
	}$, $\vect{b}=\vect{0}$
	\item $A = %
	\kgmmatrix{5}{1 & 0 & 3 & 1 & -1\\
	2 & -1 & 2 & 0 & 1\\
	-1 & 1 & 3 & 3 & 2}$, $\vect{b} = \MTBO{-1}{3}{0}$
\end{enumerate}

\item Let $A$ be an $(m\times n)$ matrix, let $\vect{v} \in \RR^n$, and let $c$ be a scalar.  Prove that $A(c\vect{v}) = c(A\vect{v})$.

\newpage
\item A game show has $n$ treasure chests arranged in a circle, indexed from $0$ through $n-1$.  For $0\le i\le n-1$, let $t_i$ be the number of dollars in the $i$th chest; these amounts are not known to the contestant.  On the line between the $i$th and $(i+1)$st treasure chest, there is a sign visible to the contestant that reveals the sum $t_i + t_{i+1}$ of the amounts in the $i$th and $(i+1)$st chests.  The object of the game is for the contestant to select the chest with maximum value.
\begin{enumerate}
    \item In this situation, which chest should the contestant pick?  Justify your answer.
    \begin{center}
    \begin{tikzpicture}[rotate=90]
        \scriptsize
        \foreach \n in {0,...,4}
        {{
            \pgfmathsetmacro{\ang}{\n*72}
            \node[circle,draw] at (\ang:2cm) (T\n) {$t_{\n}$} ;
        }}
        \foreach \n/\val in {0/11,1/12,2/12,3/11,4/20}
        {{
            \pgfmathsetmacro{\succ}{int(mod(\n + 1, 5))}
            \draw (T\n) -- node[fill=white] {$\val$} (T\succ) ;
        }} 
    \end{tikzpicture}
    \end{center}
    \item Give an example of a situation with four treasure chests where it is impossible for the contestant to know which chest to pick.
    \item Characterize the values of $n$ such that the contestant can determine the amounts $t_i$ in the $n$ chests from the given sums $t_i + t_{i+1}$, no matter what the sums may be.
\end{enumerate}

\end{enumerate}
\end{document}
