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{\large

{\sc Name} \rule{3in}{0.4pt} \hfill {\sc Score} \rule{1in}{0.4pt}
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Math 221  Introduction to Matrix Theory \\
{\bf Quiz 1} \\ 
5/23/00
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{\bf Show your work!} The quiz is worth 10 points.

\begin{enumerate}

\item Explain what it means for two vectors to be independent. For two
      vectors to be orthogonal?

\vspace{2.75in}

\item Consider the vectors ${\bf u}$ and ${\bf v}$ as drawn below.
\begin{center}\epsfig{file=vects.eps,width=2.75in}\end{center}
\begin{itemize}
\item[a)] Find the vectors ${\bf u}$ and ${\bf v}$. 
\item[b)] Find ${\bf u} +2{\bf v}$ and sketch this vector. 
\item[c)] Find a unit vector in the same direction as ${\bf u}$.
\item[d)] Find the angle between ${\bf u}$ and ${\bf v}$.
\end{itemize}

\pagebreak

\item Explain what it means for a linear system to be singular.

\vspace{2.5in} 

\item Consider the matrices
\[ 
A= \left[ \begin{array}{ccc} -1&2&3 \\ 4&1&-1 \end{array} \right],\;
B= \left[ \begin{array}{cc} 6&-1 \\ 2&4 \\ -3&5 \end{array} \right],\; 
C= \left[ \begin{array}{cc} 1&2 \\ 0&7 \end{array} \right].
\]
Find the following (if possible).
\begin{itemize}
\item[a)] $AB$
\item[b)] $AC$
\item[c)] $BC$
\item[d)] $C^{2} = CC$
\item[e)] $B+C$
\item[f)] $5A$
\item[g)] $B^{T}$
\item[h)] The matrix that is upper triangular?
\end{itemize}       

\end{enumerate}

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