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\headers{Math343}{Homework 5}{Due Sep 24, 2025}

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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 {} [1.6.6] Balance the following chemical reaction.
\[ \mathrm{Na_3PO_4 + Ba(NO_3)_2 \to Ba_3(PO_4)_2 + NaNO_3}  \]

\item {} [1.7.\{1,3,4\}] Determine if the vectors are linearly independent.  Justify your answer.
\begin{multicols}{2}
\begin{enumerate}
	\item $\MTBO{5}{1}{0}$, $\MTBO{7}{2}{-6}$, $\MTBO{-2}{-1}{6}$
	\item $\MTWBO{1}{-3}$, $\MTWBO{-3}{6}$
	\item $\MTWBO{-1}{4}$, $\MTWBO{-2}{8}$
\end{enumerate}
\end{multicols}

\item {} [1.7.\{9,10\}] For which values of $h$ is $\vect{v}_3$ in $\Span\{\vect{v}_1,\vect{v}_2\}$?  For which values of $h$ is $\{\vect{v}_1,\vect{v}_2,\vect{v}_3\}$ linearly dependent?
\begin{enumerate}
	\item $\vect{v}_1 = \MTBO{1}{-3}{2}$, $\vect{v}_2 = \MTBO{-3}{10}{-6}$, $\vect{v}_3 = \MTBO{2}{-7}{h}$
	\item $\vect{v}_1 = \MTBO{1}{-5}{3}$, $\vect{v}_2 = \MTBO{-2}{10}{6}$, $\vect{v}_3 = \MTBO{2}{-10}{h}$
\end{enumerate}

\item True/False.  Justify your answers.
\begin{enumerate}
	\item Two vectors are linearly dependent if and only if they lie on a line through the origin.
	\item If $S$ is a linearly dependent set, then each vector in $S$ is a linear combination of the other vectors in $S$.
	\item If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent.
	\item If $\vect{x}$ and $\vect{y}$ are linearly independent but $\{\vect{x},\vect{y},\vect{z}\}$ is linearly dependent, then $\vect{z}\in\Span\{\vect{x},\vect{y}\}$.
\end{enumerate}

\item Let $\vect{u_1},\ldots,\vect{u_n}$ be vectors, and suppose that $\vect{v}$ is a vector that can be obtained as a linear combination of $\vect{u_1},\ldots,\vect{u_n}$ in two different ways.  Prove that $\vect{u_1},\ldots,\vect{u_n}$ are linearly dependent.

\end{enumerate}
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