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\headers{Math343}{Homework 10}{Due Oct 29, 2025}

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\noindent
\textbf{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.

\begin{enumerate}

\item {} [2.9.5] Let $\vv b_1 =\MTBO{1}{5}{-3}$, $\vv b_2 = \MTBO{-3}{-7}{5}$, and $\vv x = \MTBO{4}{10}{-7}$.  Let $\BB$ be the basis given by the ordered set $\{\vv b_1, \vv b_2\}$.  Find the $\BB$-coordinate vector of $\vv x$, which we also denote by $[\vv x]_{\BB}$.

\item {} [2.9.\{17-22\}] True/False.  Justify your answers.
\begin{enumerate}
	\item If $\B = \{\vv v_1, \ldots, \vv v_p\}$ is a basis for a subspace $H$ and if $\vv x = c_1\vv v_1 + \cdots + c_p\vv v_p$, then $c_1,\ldots,c_p$ are the coordinates of $\vv x$ relative to the basis $\B$.
	\item 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 $\B$.
	\item Each line in $\RR^n$ is a one-dimensional subspace of $\RR^n$.
	\item The dimension of the column space of $A$ is $\rank A$.
	\item If $H$ is a $p$-dimensional subspace of $\RR^n$, then a linearly independent set of $p$ vectors in $H$ is a basis for $H$.
\end{enumerate}

\item {} [3.1.\{1-14\}] Use cofactor expansion to compute the determinants of the following matrices.
\begin{multicols}{2}
\begin{enumerate}
	\item $\mdet{3}{3 & 0 & 4 \\ 2 & 3 & 2 \\ 0 & 5 & -1}$
	\item $\mdet{3}{4 & 3 & 0 \\ 6 & 5 & 2 \\ 9 & 7 & 3}$
	\item $\mdet{4}{1 & -2 & 4 & 2\\ 0 & 0 & 3 & 0\\ 2 & -4 & -3 & 5 \\ 2 & 0 & 3 & 5}$
	\item $\mdet{5}{4 & 0 & -7 & 3 & -5 \\ 
		        0 & 0 & 2 & 0 & 0 \\ 
		        7 & 3 & -6 & 4 & -8 \\ 
		        5 & 0 & 5 & 2 & -3\\ 
		        0 & 0 & 9 & -1& 2
		        }$
\end{enumerate}
\end{multicols}

\item {} [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.
\begin{enumerate}
	\item $\mtrix{2}{a & b\\c & d}$, $\mtrix{2}{c & d\\a & b}$
	\item $\mtrix{2}{a & b\\c & d}$, $\mtrix{2}{a & b\\kc & kd}$
	\item $\mtrix{2}{a & b\\c & d}$, $\mtrix{2}{a +kc & b + kd\\ c & d}$
\end{enumerate}

\item {} [3.1.38] Let $A=\mtrix{2}{a & b\\c & d}$ and let $k$ be a scalar.  Find a formula that relates $\det kA$ to $k$ and $\det(A)$.


\end{enumerate}
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