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\begin{document}
\headers{Math343}{Quiz 6}{Oct 3, 2025}

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

\begin{enumerate}

\item\ppoints{2}{2} Decide whether the given transformation is linear.  If the transformation is is linear, give the standard matrix.  If the transformation is not linear, then explain why.
\begin{enumerate}
	\item $\mtrix{1}{ x_1 \\ x_2 } \mapsto \mtrix{1}{0 \\ |x_1 + x_2|}$
	\vfill
	
	\item $\mtrix{1}{ x_1 \\ x_2 } \mapsto \mtrix{1}{e^3 x_1 + \tan(\pi/5) x_2 \\ -x_1}$.
	\vfill
\end{enumerate} 

\item\points{1} Let $T\st \RR^n \to \RR^m$ be a linear transform, and let $\vect{v}_1, \ldots \vect{v}_p$ be vectors in $\RR^n$.  Show that if $\{\vect{v}_1, \ldots, \vect{v}_p\}$ is a linearly dependent set, then $\{T(\vect{v}_1), \ldots, T(\vect{v}_p)\}$ is linearly dependent.
\vfill

\newpage

\item\ppoints{2}{2} Suppose that $T\st \RR^2\to \RR^3$ is a linear transform, let $\vect{u} = \mtrix{1}{ -4 \\ 1 }$ and let $\vect{v} = \mtrix{1}{ 3 \\ -2}$.  We know that $T$ maps $\vect{u}$ to $\mtrix{1}{1 \\ -2 \\ 4}$ and $T$ maps $\vect{v}$ to $\mtrix{1}{2 \\ -1 \\ -1}$.
\begin{enumerate}
	\item Find the image of $2\vect{u} - \vect{v}$ under $T$.
	\vfill
	
	\item If possible, then find $T(\vect{w})$, where $\vect{w} = \mtrix{1}{1 \\ -19}$.  If not possible, then explain why not.
	\vfill
\end{enumerate}


\item\points{1} Give a simple example of a linear transformation $T\st \RR^2 \to \RR^3$ such that the range of $T$ is $\left\{\mtrix{1}{x_1 \\ x_2 \\ x_3}\st x_1 + x_2 + x_3 = 0\right\}$.
\vspace{3cm}


\end{enumerate}
\end{document}
