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..

1. [1.6.6] Balance the following chemical reaction.

   $$N{a}_{3}P{O}_4+\mathrm{Ba}{(N{O}_3)}_2\to B{a}_3{(P{O}_4)}_2+\mathrm{NaN}{O}_3$$

2. [1.7.{1,3,4}] Determine if the vectors are linearly independent. Justify your answer.

   1. $\left[\begin{array}{c}\hfill 5\\ \hfill 1\\ \hfill 0\end{array}\right]$, $\left[\begin{array}{c}\hfill 7\\ \hfill 2\\ \hfill -6\end{array}\right]$, $\left[\begin{array}{c}\hfill -2\\ \hfill -1\\ \hfill 6\end{array}\right]$
   2. $\left[\begin{array}{c}\hfill 1\\ \hfill -3\end{array}\right]$, $\left[\begin{array}{c}\hfill -3\\ \hfill 6\end{array}\right]$
   3. $\left[\begin{array}{c}\hfill -1\\ \hfill 4\end{array}\right]$, $\left[\begin{array}{c}\hfill -2\\ \hfill 8\end{array}\right]$

3. [1.7.{9,10}] For which values of $h$ is ${𝐯}_3$ in $\mathrm{Span}\{{𝐯}_1,{𝐯}_2\}$? For which values of $h$ is $\{{𝐯}_1,{𝐯}_2,{𝐯}_3\}$ linearly dependent?

   1. ${𝐯}_1=\left[\begin{array}{c}\hfill 1\\ \hfill -3\\ \hfill 2\end{array}\right]$, ${𝐯}_2=\left[\begin{array}{c}\hfill -3\\ \hfill 10\\ \hfill -6\end{array}\right]$, ${𝐯}_3=\left[\begin{array}{c}\hfill 2\\ \hfill -7\\ \hfill h\end{array}\right]$
   2. ${𝐯}_1=\left[\begin{array}{c}\hfill 1\\ \hfill -5\\ \hfill 3\end{array}\right]$, ${𝐯}_2=\left[\begin{array}{c}\hfill -2\\ \hfill 10\\ \hfill 6\end{array}\right]$, ${𝐯}_3=\left[\begin{array}{c}\hfill 2\\ \hfill -10\\ \hfill h\end{array}\right]$

4. True/False. Justify your answers.

   1. Two vectors are linearly dependent if and only if they lie on a line through the origin.
   2. If $S$ is a linearly dependent set, then each vector in $S$ is a linear combination of the other vectors in $S$.
   3. If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent.
   4. If $𝐱$ and $𝐲$ are linearly independent but $\{𝐱,𝐲,𝐳\}$ is linearly dependent, then $𝐳\in \mathrm{Span}\{𝐱,𝐲\}$.
5. Let ${𝐮}_1,\dots ,{𝐮}_{𝐧}$ be vectors, and suppose that $𝐯$ is a vector that can be obtained as a linear combination of ${𝐮}_1,\dots ,{𝐮}_{𝐧}$ in two different ways. Prove that ${𝐮}_1,\dots ,{𝐮}_{𝐧}$ are linearly dependent.