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. [2.9.5] Let ${𝐛}_1=\left[\begin{array}{c}\hfill 1\\ \hfill 5\\ \hfill -3\end{array}\right]$, ${𝐛}_2=\left[\begin{array}{c}\hfill -3\\ \hfill -7\\ \hfill 5\end{array}\right]$, and $𝐱=\left[\begin{array}{c}\hfill 4\\ \hfill 10\\ \hfill -7\end{array}\right]$. Let $B$ be the basis given by the ordered set $\{{𝐛}_1,{𝐛}_2\}$. Find the $B$-coordinate vector of $𝐱$, which we also denote by ${[𝐱]}_B$.

2. [2.9.{17-22}] True/False. Justify your answers.

   1. If $B=\{{𝐯}_1,\dots ,{𝐯}_p\}$ is a basis for a subspace $H$ and if $𝐱=c_1{𝐯}_1+\cdots +c_p{𝐯}_p$, then $c_1,\dots ,c_p$ are the coordinates of $𝐱$ relative to the basis $B$.
   2. If $B$ is a basis for a subspace $H$, then each vector in $H$ can be written in only one way as a linear combination of the vectors in $B$.
   3. Each line in ${ℝ}^n$ is a one-dimensional subspace of ${ℝ}^n$.
   4. The dimension of the column space of $A$ is $\mathrm{rank}<!--\; FUNCTION\; APPLICATION\; -->A$.
   5. If $H$ is a $p$-dimensional subspace of ${ℝ}^n$, then a linearly independent set of $p$ vectors in $H$ is a basis for $H$.

3. [3.1.{1-14}] Use cofactor expansion to compute the determinants of the following matrices.

   1. $\left|\begin{array}{ccc}\hfill 3& \hfill 0& \hfill 4\\ \hfill 2& \hfill 3& \hfill 2\\ \hfill 0& \hfill 5& \hfill -1\end{array}\right|$
   2. $\left|\begin{array}{ccc}\hfill 4& \hfill 3& \hfill 0\\ \hfill 6& \hfill 5& \hfill 2\\ \hfill 9& \hfill 7& \hfill 3\end{array}\right|$
   3. $\left|\begin{array}{cccc}\hfill 1& \hfill -2& \hfill 4& \hfill 2\\ \hfill 0& \hfill 0& \hfill 3& \hfill 0\\ \hfill 2& \hfill -4& \hfill -3& \hfill 5\\ \hfill 2& \hfill 0& \hfill 3& \hfill 5\end{array}\right|$
   4. $\left|\begin{array}{ccccc}\hfill 4& \hfill 0& \hfill -7& \hfill 3& \hfill -5\\ \hfill 0& \hfill 0& \hfill 2& \hfill 0& \hfill 0\\ \hfill 7& \hfill 3& \hfill -6& \hfill 4& \hfill -8\\ \hfill 5& \hfill 0& \hfill 5& \hfill 2& \hfill -3\\ \hfill 0& \hfill 0& \hfill 9& \hfill -1& \hfill 2\end{array}\right|$

4. [3.1.{19,20,22}] These exercises explore the effect of an elementary row operation on the determinant of a matrix. In each case, state the row operation and describe how it affects the determinant.

   1. $\left[\begin{array}{cc}\hfill a& \hfill b\\ \hfill c& \hfill d\end{array}\right]$, $\left[\begin{array}{cc}\hfill c& \hfill d\\ \hfill a& \hfill b\end{array}\right]$
   2. $\left[\begin{array}{cc}\hfill a& \hfill b\\ \hfill c& \hfill d\end{array}\right]$, $\left[\begin{array}{cc}\hfill a& \hfill b\\ \hfill \mathit{𝑘𝑐}& \hfill \mathit{𝑘𝑑}\end{array}\right]$
   3. $\left[\begin{array}{cc}\hfill a& \hfill b\\ \hfill c& \hfill d\end{array}\right]$, $\left[\begin{array}{cc}\hfill a+\mathit{𝑘𝑐}& \hfill b+\mathit{𝑘𝑑}\\ \hfill c& \hfill d\end{array}\right]$
5. [3.1.38] Let $A=\left[\begin{array}{cc}\hfill a& \hfill b\\ \hfill c& \hfill d\end{array}\right]$ and let $k$ be a scalar. Find a formula that relates $\det <!--\; FUNCTION\; APPLICATION\; -->\mathit{𝑘𝐴}$ to $k$ and $\det <!--\; FUNCTION\; APPLICATION\; -->(A)$.