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EXAM 2 &\huge TQS 308  &\normalsize Fall 2010\\
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\vskip .5 in
NAME:


\noindent{True/False: If the statement is \emph{always} true, give a brief explanation of why it is.  If the statement is false, give a counterexample.}


\normalsize
\begin{enumerate}
\item $[4]$  If $A$ and $B$ are both $2\times 2$ matrices, then $AB=BA$
\vfill
\item $[4]$ A matrix of the form $\left[ \begin{matrix}
a &0\\
0 &b\\
\end{matrix}\right]$ where $a$, $b \in \mathbb{R}$ commutes with $\left[ \begin{matrix}
1 &0\\
0 &0\\
\end{matrix}\right]$ 
\vfill
\item $[4]$ If $A$ and $B$ are both $2\times 2$ matrices, then $(A+B)^{-1}=A^{-1}+B^{-1}$.
\vfill

\newpage
\item $[4]$ A basis for the subspace: $\left\{\left[ \begin{matrix}
1\\
1\\
1\\
\end{matrix}\right]
+t\left[ \begin{matrix}
1\\
2\\
3\\
\end{matrix}\right]
+r\left[ \begin{matrix}
3\\
6\\
9\\
\end{matrix}\right] \vert r,t\in \mathbb{R} \right\}
$ in $\mathbb{R}^3$ is: $\left\{ \left[ \begin{matrix}
1\\
2\\
3\\
\end{matrix}\right] \text{, }
\left[ \begin{matrix}
3\\
6\\
9\\
\end{matrix}\right] \right\}$
\vfill
\item $[4]$ A set of vectors that are linearly independent in $\mathbb{R}^3$ also form a basis for $\mathbb{R}^3$.
\vfill
\item $[4]$ The transformation $T:\mathbb{R}^2\rightarrow \mathbb{R}^3$ that embeds $\mathbb{R}^2$ into the $xy$-plane is given by 
$$\overrightarrow{x} \longrightarrow \left[ \begin{matrix}
1&0&0\\
0&1&0\\
\end{matrix}\right]\overrightarrow{x}.$$

\vfill


\newpage
\large
Free Responce: Show your work for the following problems.  The correct answer with 
no supporting work will receive NO credit.

\normalsize
\item Let $A$, $B$, and $C$ be defined by:

\hspace{1in}$A= \left[  \begin{matrix}
   				   	-2 & 2 & 0\\
      					4 & -4 & 1 \\
   			\end{matrix}\right]$,
$B= \left[  \begin{matrix}
   				   	-3\\
      					6 \\
					9\\
   			\end{matrix}\right]$, and 
$C= \left[  \begin{matrix}
   				   	-2 & 2 & 0\\
					0 &1 &-1\\
					0 &0 &1\\
   			\end{matrix}\right].$

\noindent Compute the following if possible. 


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(a) $[1]$ $2B$ & (b) $[2]$ $A^\text{T}$\\
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\vskip 1in

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(c) $[2]$ a basis for col($A$) &(d) $[4]$ $C^{-1}$\\
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\vfill




\item Let $A$ be a $2\times 3$ matrix, $B$ be a $3\times 1$ matrix, and let $C$ be a $3\times 3$ matrix.
\begin{enumerate}
\item $[1]$ What are the dimensions of the matrix $BC$, if it exists?
\vskip .5in
\item $[1]$ What are the dimensions of the matrix $CA^\text{T}$, if it exists?
\vskip .5in
\item $[1]$ What are the dimensions of the matrix $C^8B$, if it exists?
\vskip .5in
\end{enumerate}

\newpage
\item Assume all matrices $A_i$ given below are multiplicatively invertible and of such a size that the multiplication makes sense.
\begin{enumerate} 
\item $[2]$ What is the multiplicative inverse of $A_2A_1$?
\vskip 1in
\item $[4]$ Use induction to prove that $A_n..A_2A_1$ is invertible for all $n$. 
\end{enumerate}




\newpage
\item  Consider the transformation $T:\mathbb{R}^2\rightarrow \mathbb{R}^2$ defined by  $T\overrightarrow{v}=A\overrightarrow{v}$, where 

\hspace{1in}$A= \left[  \begin{matrix}
   				   	\frac{1}{2} &  -\frac{1}{2}\\
					-\frac{1}{2} & \frac{1}{2}\\
   			\end{matrix}\right]$.
\begin{enumerate}
\item $[1]$ The vector shown on the right is $\overrightarrow{v}$.\\
Draw the image of $\overrightarrow{v}$ under the \\
transformation $T$.

\vspace{-1in} \hspace{3in}\includegraphics[height=2.5in]{vector1}	

\item $[3]$ Find a basis for $null(A)$.
\vskip 2in
\item $[2]$ Identify $null(A)$ on the axes provided above.
\vskip .2in
\item $[2]$ Find any $\overrightarrow{x}$ that are \emph{invariant} under the action of $T$.  That is, find all $\overrightarrow{x}$ such that $T\overrightarrow{x}=\overrightarrow{x}$.  Justify your answer either geometrically or algebraically.

\end{enumerate}







\end{enumerate}

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