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. [1.3.{11,12}] Determine if $𝐛$ is a linear combination of ${𝐚}_1$, ${𝐚}_2$, and ${𝐚}_3$.

   1. ${𝐚}_1=\left[\begin{array}{c}\hfill 1\\ \hfill -2\\ \hfill 0\end{array}\right]$, ${𝐚}_2=\left[\begin{array}{c}\hfill 0\\ \hfill 1\\ \hfill 2\end{array}\right]$, ${𝐚}_3=\left[\begin{array}{c}\hfill 5\\ \hfill -6\\ \hfill 8\end{array}\right]$, $𝐛=\left[\begin{array}{c}\hfill 2\\ \hfill -1\\ \hfill 6\end{array}\right]$.
   2. ${𝐚}_1=\left[\begin{array}{c}\hfill 1\\ \hfill -2\\ \hfill 2\end{array}\right]$, ${𝐚}_2=\left[\begin{array}{c}\hfill 0\\ \hfill 5\\ \hfill 5\end{array}\right]$, ${𝐚}_3=\left[\begin{array}{c}\hfill 2\\ \hfill 0\\ \hfill 8\end{array}\right]$, $𝐛=\left[\begin{array}{c}\hfill -5\\ \hfill 11\\ \hfill -7\end{array}\right]$.

2. [1.3.13] Determine if $𝐛$ is a linear combination of the vectors formed from the columns of the matrix $A$.

   $$\begin{array}{llllllll}\hfill A& =\left[\begin{array}{ccc}\hfill 1& \hfill -4& \hfill 2\\ \hfill 0& \hfill 3& \hfill 5\\ \hfill -2& \hfill 8& \hfill -4\end{array}\right]& \hfill 𝐛& =\left[\begin{array}{c}\hfill 3\\ \hfill -7\\ \hfill -3\end{array}\right]& \hfill & & \hfill & \end{array}$$

3. [1.3] True/False. Justify your answers.

   1. Another notation for the vector $\left[\begin{array}{c}\hfill -4\\ \hfill 3\end{array}\right]$ is $[-4 3]$.
   2. The points in the plane corresponding to $\left[\begin{array}{c}\hfill -2\\ \hfill 5\end{array}\right]$ and $\left[\begin{array}{c}\hfill -5\\ \hfill 2\end{array}\right]$ lie on a line through the origin.
   3. An example of a linear combination of vectors ${𝐯}_1$ and ${𝐯}_2$ is the vector $\frac{1}{2}{𝐯}_1$.
   4. The weights $c_1,\dots ,c_p$ in a linear combination $c_1{𝐯}_1+\cdots +c_p{𝐯}_{𝐩}$ cannot all be zero.
   5. The set $\mathrm{Span}\{𝐮,𝐯\}$ is always visualized as a plane through the origin.

4. Pivot columns in a matrix.

   1. Let $A=[ {𝐚}_1 {𝐚}_2 \cdots {𝐚}_{𝐩} ]$ and let $B=[ {𝐛}_1 {𝐛}_2 \cdots {𝐛}_{𝐩} ]$. Suppose that ${𝐚}_{𝐢}={𝐚}_{𝐣}$ and $A$ is row equivalent to $B$. Explain why it must be that ${𝐛}_{𝐢}={𝐛}_{𝐣}$. (Hint: how do the $i$th and $j$th columns behave with respect to each elementary row operation?)
   2. Let $A=[ {𝐚}_1 {𝐚}_2 \cdots {𝐚}_{𝐩} ]$. Suppose that ${𝐚}_{𝐢}={𝐚}_{𝐣}$ and that $i<j$. Use part (a) to explain why the $j$th column of $A$ cannot be a pivot column. (Hint: let $B$ be the reduced echelon form of $A$.)

5. [1.4.9] Write the system first as a vector equation and then as a matrix equation.

   $$\begin{array}{ccccccc}\hfill 3x_1& \hfill +& \hfill x_2& \hfill -& \hfill 5x_3& \hfill =& \hfill 9\\ \hfill & \hfill & \hfill x_2& \hfill +& \hfill 4x_3& \hfill =& \hfill 0\end{array}$$

6. [1.4.11] Given $A$ and $𝐛$ below, write augmented matrix corresponding to the matrix equation $A𝐱=𝐛$ and solve for $𝐱$.

   $$\begin{array}{llllllll}\hfill A& =\left[\begin{array}{ccc}\hfill 1& \hfill 2& \hfill 4\\ \hfill 0& \hfill 1& \hfill 5\\ \hfill -2& \hfill -4& \hfill -3\end{array}\right]& \hfill B& =\left[\begin{array}{c}\hfill -2\\ \hfill 2\\ \hfill 9\end{array}\right]& \hfill & & \hfill & \end{array}$$