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$T \ \mbox{is linear, since} \ T({\bf 0})={\bf 0}$

$\mbox{None of the above}$

$T \ \mbox{is linear, since} \ T({\bf x})=A{\bf x}\ \mbox{where}\ A=\left [\begin{array}{rrr}1&0&-1\\[3mm]0&0&5\\[3mm]-2&1&0\end{array}\right ]$

$T \ \mbox{is linear, since} \ T({\bf x})=A{\bf x}\ \mbox{where}\ A=\left [\begin{array}{rrr}1&0&-2\\[3mm]0&0&1\\[3mm]-1&5&0\end{array}\right ]$

$T \, \mbox{is not linear, since} \ T(c{\bf u})\not =cT({\bf u})$

$T \ \mbox{is not linear, since} \ T({\bf u}+{\bf v})\not =T({\bf u})+T({\bf v})$

$T \, \mbox{is not linear, since} \ T({\bf 0})\not ={\bf 0}$

	$\hspace{-6mm} V \ \mbox{is not a vector space, since there exist} \ {\bf u} \ \mbox{and} \ {\bf v} \ \mbox{from} \ V\\ \mbox{such that} \ {\bf u}+{\bf v}\ \mbox{is not from} \ V$
	$V \ \mbox{is not a vector space, since} \ {\bf 0} \ \mbox{is not in} \ V$
	$V \ \mbox{is a vector space, since} \ {\bf 0} \ \mbox{is in} \ V$
	$V \ \mbox{is a vector space, since} \ c{\bf u}\ \mbox{is from} \ V \ \mbox{for any} \ {\bf u}\ \mbox{from} \ V \ \mbox{and any} \ c \ \mbox{from} \ \mathbb{R}$
	$\mbox{None of the above}$
	$V \ \mbox{is a vector space, since} \ {\bf u}+{\bf v}\ \mbox{is from} \ V \ \mbox{for any} \ {\bf u}\ \mbox{and} \ {\bf v} \ \mbox{from} \ V$

Test!

	$A^{-1}=\left [\begin{array}{rrr}3&-7&3\\[3mm]-1&1&-2\\[3mm]1&-2&-1\end{array}\right ]$
	$A^{-1}=\left [\begin{array}{rrr}3&-1&1\\[3mm]-7&1&-2\\[3mm]3&-2&-1\end{array}\right ]$
	$A^{-1}=\left [\begin{array}{rrr}3&-1&1\\[3mm]-7&5&-2\\[3mm]3&-2&1\end{array}\right ]$
	$A^{-1} =\left [\begin{array}{rrr}3&-7&3\\[3mm]-1&5&-2\\[3mm]1&-2&1\end{array}\right ]$

	${\bf x}(t)=c_1e^{-2t}\left [\begin{array}{c}1\\[1mm]0\\[1mm]0\end{array}\right ]+c_2e^{2t}\left [\begin{array}{c}0\\[1mm]1\\[1mm]1\end{array}\right ]+c_3e^{3t}\left [\begin{array}{c}0\\[1mm]7\\[1mm]2\end{array}\right ]$
	${\bf x}(t) =c_1e^{-2t}\left [\begin{array}{c}0\\[1mm]1\\[1mm]1\end{array}\right ]+c_2e^{2t}\left [\begin{array}{c}1\\[1mm]0\\[1mm]0\end{array}\right ]+c_3e^{3t}\left [\begin{array}{c}0\\[1mm]7\\[1mm]2\end{array}\right ]$
	$\mbox{None of the above}$
	${\bf x}(t)=c_1e^{-2t}\left [\begin{array}{c}0\\[1mm]1\\[1mm]1\end{array}\right ]+c_2e^{2t}\left [\begin{array}{c}1\\[1mm]0\\[1mm]0\end{array}\right ]+c_3e^{-3t}\left [\begin{array}{c}0\\[1mm]7\\[1mm]2\end{array}\right ]$
	${\bf x}(t)=c_1e^{-2t}\left [\begin{array}{c}1\\[1mm]0\\[1mm]1\end{array}\right ]+c_2e^{2t}\left [\begin{array}{c}0\\[1mm]1\\[1mm]0\end{array}\right ]+c_3e^{3t}\left [\begin{array}{c}0\\[1mm]7\\[1mm]2\end{array}\right ]$
	${\bf x}(t)=c_1e^{-2t}\left [\begin{array}{c}1\\[1mm]1\\[1mm]0\end{array}\right ]+c_2e^{2t}\left [\begin{array}{c}1\\[1mm]0\\[1mm]1\end{array}\right ]+c_3e^{3t}\left [\begin{array}{c}0\\[1mm]2\\[1mm]7\end{array}\right ]$

	${\bf x}(t)=c_1e^{5t}\left [\begin{array}{c}1\\[1mm]0\\[1mm]0\end{array}\right ]+c_2\left [\begin{array}{c}\cos t\\[1mm]-\sin t\\[1mm]0\end{array}\right ]+c_3\left [\begin{array}{c}\sin t\\[1mm]\cos t\\[1mm]0\end{array}\right ]$
	${\bf x}(t)= c_1e^{5t}\left [\begin{array}{c}0\\[1mm]0\\[1mm]1\end{array}\right ]+c_2\left [\begin{array}{c}\cos t\\[1mm]-\sin t\\[1mm]0\end{array}\right ]+c_3\left [\begin{array}{c}\sin t\\[1mm]\cos t\\[1mm]0\end{array}\right ]$
	${\bf x}(t)= c_1e^{5t}\left [\begin{array}{c}0\\[1mm]0\\[1mm]1\end{array}\right ]+c_2\left [\begin{array}{c}\cos t\\[1mm]-\sin t\\[1mm]0\end{array}\right ]+c_3\left [\begin{array}{c}-\sin t\\[1mm]\cos t\\[1mm]0\end{array}\right ]$
	${\bf x}(t)=c_1e^{5t}\left [\begin{array}{c}0\\[1mm]0\\[1mm]1\end{array}\right ]+c_2\left [\begin{array}{c}\cos t\\[1mm]-\sin t\\[1mm]0\end{array}\right ]+c_3\left [\begin{array}{c}\sin t\\[1mm]0\\[1mm]\cos t\end{array}\right ]$
	$\mbox{None of the above}$
	${\bf x}(t)=c_1e^{5t}\left [\begin{array}{c}0\\[1mm]0\\[1mm]1\end{array}\right ]+c_2\left [\begin{array}{c}\cos t\\[1mm]\sin t\\[1mm]0\end{array}\right ]+c_3\left [\begin{array}{c}\sin t\\[1mm]\cos t\\[1mm]0\end{array}\right ]$

	$\mbox{Independent}$
	$\mbox{Dependent}$

	$\mbox{No }$
	$\mbox{Yes}$