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\begin{document}
\headers{Math343}{Quiz 10}{Oct 31, 2025}

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\noindent
\textbf{Directions:} Show all work.  No credit for answers without work.

\begin{enumerate}

\item\points{2} Let $\B = \{\vect{b}_1, \vect{b}_2, \mb b_3\}$, where $\vect{b}_1 = \mtrix{1}{5 \\ -1 \\ 4 \\ 2}$, $\vect{b}_2 = \mtrix{1}{-2 \\ 3 \\ 1 \\ 1}$, and $\mb b_3 = \mtrix{1}{2 \\ 0 \\ 3 \\ 1}$.  Given $\vect{x}=\mtrix{1}{3 \\ 12 \\ 14 \\ 10}$, find $[\vect{x}]_{\B}$ if possible.
\vfill

\item\points{2} Let $A$ be a $5\times 8$ matrix.  
\begin{enumerate}
	\item What are the possible values for the dimension of the null space of $A$?
	\vspace{3cm}
	\item Suppose that the transform $T$ given by $T(\mb x) = A\mb x$ is onto/surjective.  Now what are the possible values for the dimension of the null space of $A$?
	\vspace{3cm}
\end{enumerate}

\newpage

\item Compute the determinant of the following matrices.
\begin{enumerate}
	\item\points{1} $\mtrix{2}{3 & -1 \\ 2 & 3}$
	\vspace{3cm}
	\item\points{1} $\mtrix{3}{2 & -1 & 5 \\ 0 & 1 & -2 \\ 1 & 7 & -2}$
	\vspace{3cm}
	\item\points{2} $\mtrix{4}{2 & 6 & 0 & -5 \\ 4 & 1 & 3 & 8 \\ 0 & 5 & 0 & 0 \\ 3 & -2 & 0 & 1}$
	\vspace{4cm}
	\item\points{2} $\mtrix{4}{1 & 3 & 1 & 5 \\ -2 & -4 & -3 & -6 \\ 1 & 3 & 2 & 8 \\ 3 & 9 & 3 & 12}$  (Hint: use row reduction)
	\vfill
\end{enumerate}

\end{enumerate}
\end{document}
