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\title{314 Spring '09-QUIZ 5}
\author{Name:\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ }
\maketitle
\section{$< 10pts >$}
Let ${\bf x} = (1,-1,1)^T$, ${\bf y} = (-1,1,-2)^T$. Compute
$||{\bf x} - {\bf y}||_1$, $||{\bf x} - {\bf y}||_2$, 
$||{\bf x} - {\bf y}||_\infty $. Under which norm are the two
vectors closest together? Under which norm are they farthest 
apart?

{\bf Solution}\\
\ \    

\newpage
\section{$< 10pts >$}
Find the vector projection ${\bf p}$ of ${\bf x}$ onto ${\bf y}$
and verify that ${\bf p}$ and ${\bf x} - {\bf p}$ are
orthogonal if 
${\bf x} = (1,-1,1)^T$, ${\bf y} = (-1,1,-2)^T$.

{\bf Solution}\\
\ \
                                                                                
                                                                                
                                                                                

\newpage
\section{$< 20pts >$}
Consider the matrix
\[ {\bf A} = \left(
\begin{array}{rrr}
1 & 0 & 2 \\-1 & 0 & 0 \\1 & 1 & 0\\1 & 1 & -2
\end{array}\right)   \]
\begin{enumerate}
\item Use the Gram-Schmidt process to construct the QR factorization.
\item Use the QR factorization found in part (1) to solve the least
squares problem
$ A {\bf x} = (1, -1, 0, 1)^T$.
\end{enumerate}

{\bf Solution}\\
\ \ 



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