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EXAM 2 &\huge TMath 308  &\normalsize Fall 2013\\
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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 $[80]$  The vector $\overrightarrow{v}=\left[ \begin{matrix}
1\\
2\\
3\\
\end{matrix}\right]$ is in the span$\left(\left[ \begin{matrix}
1\\
1\\
0\\
\end{matrix}\right],
\left[ \begin{matrix}
0\\
1\\
1\\
\end{matrix}\right],
\left[ \begin{matrix}
1\\
0\\
1\\
\end{matrix}\right]\right)$.

\vfill
\item $[80]$ Given matrices $A$, $B$, and $C$ and that $A*B=A*C$, then $B=C$.


\vfill
\item $[80]$ If $A$ and $B$ are both $2\times 2$ matrices, then $(A+B)^{-1}=A^{-1}+B^{-1}$.
\vfill

\newpage
\item $[80]$ The set  $\left\{\left[\begin{matrix}
1\\
1\\
0\\
\end{matrix}\right],
\left[ \begin{matrix}
0\\
1\\
1\\
\end{matrix}\right],
\left[ \begin{matrix}
1\\
0\\
1\\
\end{matrix}\right]\right\}$ is linearly independent.


\vfill
\item $[80]$  If $A$ and $B$ are symmetric $n\times n$ matrices, then $AB$ is symmetric.
\vfill
\item $[80]$ The set $\{(x,y,z)\in \mathbb{R}^3 \vert x-y+z=1\}$ is a subspace.
\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}$A= \left[  \begin{matrix}
   				   	3 & 0 & 0\\
      					-1 & 5 & 1 \\
   			\end{matrix}\right]$,
$B= \left[  \begin{matrix}
   				   	4 &1\\
      					0  & 0\\
					3 & 0\\
   			\end{matrix}\right]$, 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) $[60]$ $A+B^\text{T}$ & (b) $[40]$ $CA$\\
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(c) $[60]$ a basis for col($A$) &(d) $[80]$ $C^{-1}$\\
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\vfill





\newpage

\item $[60]$ Given that $A$, $B$, and $X$ are all matrices of appropriate sizes, solve for the matrix $X$.  Assume all matrices are invertible and simplify as much as possible. $$ABXA^{-1}B^{-1}=I+A$$

\vspace{2in}

\item $[80]$ Prove that if $AB$ and $BA$ are both defined then $AB$ and $BA$ are both square matrices.






\newpage
\item  Let $W$ be the set of all vectors $\overrightarrow{u}\in \mathbb{R}^2$ that exhibit the property:
$$\vert\vert \overrightarrow{u}+\left[ \begin{array}{c}-1\\1\\\end{array}\right] \vert\vert=\vert\vert\overrightarrow{u}\vert\vert+\vert\vert\left[ \begin{array}{c}-1\\1\\\end{array}\right]\vert\vert.$$
\noindent In set builder notation: 
$$W=\left\{ \overrightarrow{u}\in \mathbb{R}^2 ;  \vert\vert \overrightarrow{u}+\left[ \begin{array}{c}-1\\1\\\end{array}\right] \vert\vert=\vert\vert\overrightarrow{u}\vert\vert+\vert\vert\left[ \begin{array}{c}-1\\1\\\end{array}\right]\vert\vert\right\}.$$


\begin{enumerate}
\item $[60]$ Describe geometrically what vectors the set $W$ contains.  (A picture is worth a thousand words and no proof is needed.)

\item $[80]$ Either prove $W$ is a subspace or determine which subspace property is not satisfied by $W$.

\end{enumerate}











\end{enumerate}

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