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\begin{document}
\title{314, S'07-QUIZ 3}
\author{Name:\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ }
\maketitle
\section{$< 13pts >$}
Given the vectors
\[ 
{\bf x}_1 =
\left[
\begin{array}{r} 1 \\ -2 \\ 4 \end{array}
\right] \ ,\ 
{\bf x}_2 =
\left[
\begin{array}{r} -1 \\  2 \\ 4 \end{array}
\right] \ ,\ 
{\bf x}_3 =
\left[
\begin{array}{r} -1 \\  2 \\ 12 \end{array}
\right] 
\]
Give the dimension of Span$({\bf x}_1,{\bf x}_2,{\bf x}_3 )$ and find
a basis. \\[.1in]
{\bf Solution}\\

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\newpage

\section{$< 12pts >$}
Given 
\[
{\bf v}_1 =
\left[ \begin{array}{r}
       2 \\ 3 
       \end{array} \right] \ ,\ 
{\bf v}_2 =
\left[ \begin{array}{r}
      -1 \\ 2 
       \end{array} \right] \ ,\ 
S = 
\left[ \begin{array}{rr}
       3 & -1 \\ 1 & 3
       \end{array} \right] \ .
\]
Find vectors ${\bf u}_1\ ,\ {\bf u}_2$ so that S is the
transition matrix from $\left[ {\bf u}_1\ ,\ {\bf u}_2 \right]$
to $\left[ {\bf v}_1\ ,\ {\bf v}_2 \right]$. \\

\noindent {\bf Solution}\\

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\newpage

\section{$< 15pts >$}
Find a basis of the row space, column space and null space of
\[ A = 
\left[
\begin{array}{rrrrr}
 1 & -1 & \ 0 &  1 &  2 \\
 2 & -1 & \ 1 &  3 &  1 \\
-1 &  2 & \ 1 & -1 & -3
\end{array}
\right]
\]
and give the dimension of each.\\

\noindent {\bf Solution} (use reduced row echelon form to
find null vectors most easily)\\

\end{document}
