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

1. [2 points] Let $B=\{{𝐛}_1,{𝐛}_2,{𝐛}_3\}$, where ${𝐛}_1=\left[\begin{array}{c}\hfill 5\\ \hfill -1\\ \hfill 4\\ \hfill 2\end{array}\right]$, ${𝐛}_2=\left[\begin{array}{c}\hfill -2\\ \hfill 3\\ \hfill 1\\ \hfill 1\end{array}\right]$, and ${𝐛}_3=\left[\begin{array}{c}\hfill 2\\ \hfill 0\\ \hfill 3\\ \hfill 1\end{array}\right]$. Given $𝐱=\left[\begin{array}{c}\hfill 3\\ \hfill 12\\ \hfill 14\\ \hfill 10\end{array}\right]$, find ${[𝐱]}_B$ if possible.

2. [2 points] Let $A$ be a $5\times 8$ matrix.

   1. What are the possible values for the dimension of the null space of $A$?
   2. Suppose that the transform $T$ given by $T(𝐱)=A𝐱$ is onto/surjective. Now what are the possible values for the dimension of the null space of $A$?

3. Compute the determinant of the following matrices.

   1. [1 point] $\left[\begin{array}{cc}\hfill 3& \hfill -1\\ \hfill 2& \hfill 3\end{array}\right]$
   2. [1 point] $\left[\begin{array}{ccc}\hfill 2& \hfill -1& \hfill 5\\ \hfill 0& \hfill 1& \hfill -2\\ \hfill 1& \hfill 7& \hfill -2\end{array}\right]$
   3. [2 points] $\left[\begin{array}{cccc}\hfill 2& \hfill 6& \hfill 0& \hfill -5\\ \hfill 4& \hfill 1& \hfill 3& \hfill 8\\ \hfill 0& \hfill 5& \hfill 0& \hfill 0\\ \hfill 3& \hfill -2& \hfill 0& \hfill 1\end{array}\right]$
   4. [2 points] $\left[\begin{array}{cccc}\hfill 1& \hfill 3& \hfill 1& \hfill 5\\ \hfill -2& \hfill -4& \hfill -3& \hfill -6\\ \hfill 1& \hfill 3& \hfill 2& \hfill 8\\ \hfill 3& \hfill 9& \hfill 3& \hfill 12\end{array}\right]$ (Hint: use row reduction)