% m211og59.tex (prev.: oh58) - The Gram-Schmidt Orthogonalization Procedure
%  - revion of m211oh59
%
\newcommand{\titlesize}{30pt}  % for title font
%
\input{m211oh.tex}
%
\begin{titlepage}
%\vspace*{-25pt}
\begin{center}\tcr{\rmH The Gram-Schmidt\vspace{10pt}\\ Orthogonalization Procedure} \end{center} %\vspace{0pt}
%\LARGE              
\huge

\begin{description}  
 
\item[\tcr{Theorem 10}] \tcg{(Gram-Schmidt):}
Let  \tcb{$\vec{v}_{\scr1}, \ldots, \vec{v}_{\scr k}$} be any  \tcr{basis} of \tcb{$V \subset \R^n$}, and put
\tcb{
\beqn
\vec{b}_{\scr1} & = & \vec{v}_{\scr1} \\
\vec{b}_{\scr2} & = & \vec{v}_{\scr2} - \;\frac{\vec{v}_{\scr2}\cdot\vec{b}_{\scr1}}
{\vec{b}_{\scr1}\cdot\vec{b}_{\scr1}}\vec{b}_{\scr1} 
\hspace*{35pt} = \;\; \vec{v}_{\scr2} - P_{V_{1}}(\vec{v}_{\scr2})\\
\vdots\hspace*{10pt} & \vdots & \hspace*{90pt}\vdots \hspace*{85pt} \vdots \hspace{90pt} \vdots\\
\vec{b}_{\scr k} & = & \vec{v}_{\scr k} - \sum _{\scr i=1}^{\scr k-1} \frac{\vec{v}_{\scr k}\cdot\vec{b}_{\scr i}}{\vec{b}_{\scr i}\cdot\vec{b}_{\scr i}}\vec{b}_{\scr i} 
\; = \; \vec{v}_{\scr k} - P_{V_{k-1}}(\vec{v}_{\scr k}),
\eeqn}
where \tcb{$V_{\scr i} = \langle \vec{v}_{\scr1}, \ldots, \vec{v}_{\scr i}\rangle = \langle \vec{b}_{\scr1}, \ldots, \vec{b}_{\scr i}\rangle$}, if 
\tcb{$1 \le i \le k$}.  Then:

(a) \tcb{$\vec{b}_{\scr1}, \ldots, \vec{b}_{\scr k}$} is an \tcr{orthogonal} basis of \tcb{$V$}, and 

(b) \tcb{$\frac{\vec{b}_{\scr1}}{\|\vec{b}_{\scr1}\|}, \ldots, \frac{\vec{b}_{\scr k}}{\|\vec{b}_{\scr k}\|} $} is an \tcr{orthonormal} basis of \tcb{$V$}.
\end{description}
\end{titlepage}
\end{document}                          