quiz4.tex

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\Large
\rm{Math 221-AB1:  Quiz 4}
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\begin{enumerate}

\item{Let $U= \Span\{x_1, x_2\}$ where $x_1 =\left[ \begin{array}{r} 1 \\ 0 \\ 2 \\ 2 \end{array} \right]$ and 
$x_2 =\left[ \begin{array}{r} -2 \\ 1 \\ 0 \\ -1 \end{array} \right]$


\begin{enumerate}
\item{(\emph{4 pts}) Apply the Gram-Schmidt process to the basis $\{x_1,x_2\}$ to produce an orthonormal basis $\{u_1,u_2\}$ for $U$.}
\item{(\emph{3 pts}) Find any basis for $U^{\perp}$.}
\end{enumerate}

}

\vspace{4.5in}


\item{(\emph{3 pts}) Explain why $x^TAy$ is not an inner product for $x,y$ in $\mathbb{R}^2$ if 
	$A = \left[ \begin{array}{rr} 1 & 2\\ 3 & 4 \end{array}\right]$.

\vspace{0.1in}

(\textit{Hint:  Show that $x^TAy$ fails to satisfy at least one property of inner products for a specific $x$ and $y$.})
}



%\item{(\emph{3 pts}) Let $W= \Span\left\{			
%	\left[ \begin{array}{r} 1 \\ 0 \\ 3 \end{array}\right]
%	\left[ \begin{array}{r} -3 \\ 1 \\ 1 \end{array}\right]
%	\right\}$ and
%	$x = \left[ \begin{array}{r} 2 \\ 1 \\ 2 \end{array}\right]$. Calculate $a$ and $b$ so that
%
%\vspace{0.2in}
%$\Proj_W(x) = a\left[ \begin{array}{r} 1 \\ 0 \\ 3 \end{array}\right]
%+b\left[ \begin{array}{r} -3 \\ 1 \\ 1 \end{array}\right] $.  
%}


\end{enumerate}

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\end{document}