Name:      

Directions: Show all work. No credit for answers without work.

1. [3 parts, 1 point each] True/False. Justify your answers.

   1. There is a vector $𝐮\in {ℝ}^n$ such that for each vector $𝐯\in {ℝ}^n$, we have $𝐮\in \mathrm{Span}\{𝐯\}$.
   2. If $𝐮$ and $𝐯$ are both linear combinations of the vectors ${𝐚}_1,\dots ,{𝐚}_p$, then so is $𝐮+𝐯$.
   3. For all vectors $𝐮$ and $𝐯$, we have that $\mathrm{Span}\{𝐮\}=\mathrm{Span}\{𝐯\}$ implies $𝐮=𝐯$.

2. [1 point] Determine if $𝐛$ is in the span of ${𝐚}_1,{𝐚}_2,{𝐚}_3$ where ${𝐚}_1=\left[\begin{array}{c}\hfill -1\\ \hfill -3\\ \hfill 2\end{array}\right]$, ${𝐚}_2=\left[\begin{array}{c}\hfill 1\\ \hfill 4\\ \hfill 5\end{array}\right]$, ${𝐚}_3=\left[\begin{array}{c}\hfill 1\\ \hfill 6\\ \hfill 19\end{array}\right]$, and $𝐛=\left[\begin{array}{c}\hfill 1\\ \hfill 0\\ \hfill 0\end{array}\right]$.

3. [2 parts, 3 points each] Given $A$ and $𝐛$ below, solve for $𝐱$ in the matrix equation $A𝐱=𝐛$.

   1. $A=\left[\begin{array}{ccc}\hfill 5& \hfill 3& \hfill -1\\ \hfill 0& \hfill 1& \hfill -2\\ \hfill -2& \hfill 1& \hfill 1\end{array}\right]$, $𝐛=\left[\begin{array}{c}\hfill 31\\ \hfill -8\\ \hfill -15\end{array}\right]$.
   2. $A=\left[\begin{array}{ccc}\hfill 2& \hfill -1& \hfill 2\\ \hfill 1& \hfill 0& \hfill 2\\ \hfill 3& \hfill -1& \hfill -1\\ \hfill 4& \hfill 1& \hfill 0\end{array}\right]$, $𝐛=\left[\begin{array}{c}\hfill 23\\ \hfill 13\\ \hfill 11\\ \hfill 5\end{array}\right]$.