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\begin{document}
\headers{Math343}{Quiz 9}{Oct 24, 2025}

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

\begin{enumerate}

\item\points{3} Determine if the following vectors form a basis for $\RR^4$.
\[ 
\mtrix{1}{
	-3 \\
	1 \\
	0 \\
	-2
},
\mtrix{1}{
	6 \\
	-4 \\
	-1 \\
	6
},
\mtrix{1}{
	1 \\
	-6 \\
	-3 \\
	6
},
\mtrix{1}{
	-8 \\
	1 \\
	0 \\
	-2
}\]
\vfill

\item\points{3} Given the matrix $A$ and an echelon form of $A$, find a basis for $\Col(A)$ and $\Nul(A)$.
\[ A = \mtrix{5}{
	75	& 383	& -281	& 32	& 329 \\
	10	& 51	& -37	& 4	    & 44 \\
	38	& 194	& -142	& 16	& 167
} \sim
\mtrix{5}{
	1	& 5	& -3	& 0	& 6 \\
	0	& 1	& -7	& 4	& -1 \\
	0	& 0	& 0	& 0	& 1
}
\]
\vfill

\newpage
\item\points{2} Let $H$ be the set of vectors $\mtrix{1}{x_1 \\ x_2 \\ x_3 \\ x_4} \in \RR^4$ such that $x_1 + x_2 + x_3 + x_4 = 0$ and $x_1 + 2x_2 + 3x_3 + 4x_4 = 0$.  Find a basis for $H$.
\vfill

\item\ppoints{2}{1} Subspaces.
\begin{enumerate}
	\item Let $H_1$ and $H_2$ be subspaces of $\RR^n$.  Prove that $H_1\cap H_2$ is also a subspace of $\RR^n$.
	\vfill
	
	\item Give an example of subspaces $H_1$ and $H_2$ of $\RR^2$ such that $H_1 \cup H_2$ is not a subspace.
	\vspace{3cm}
\end{enumerate}

\end{enumerate}
\end{document}
