% m211oh59.tex  -  Orthogonal Vectors and Orthogonal Projection (new 2018)
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\begin{center}\tcr{\rmH Orthogonal Vectors and Orthogonal\\[2mm] Projection} \end{center} %\vspace{0pt}
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\newcommand{\vv}{\vec{v}}
\newcommand{\vb}{\vec{b}}
\newcommand{\cB}{{\cal B}} 
\begin{description}  
\item[\tcr{Definition:}] A set \mc{\vv_1, \ldots, \vv_k} of vectors of a subspace \mc{V \subset \R^n} 
is called a \tcr{\emph{basis}} of \mc{V} if

(i) the vectors \mc{\vv_1, \ldots,\vv_k} are \tcr{linearly independent;}

(ii) the vectors \mc{\vv_1, \ldots,\vv_k} \tcr{span} \mc{V}, i.e., 
\mc{V =  \lb\vv_1, \ldots,\vv_k\rb}. 

\item[\tcr{Fact:}] Each subspace \mc{V \subset \R^n} has a basis, and all bases of \mc{V} 
have the same number of elements, called the \emph{\tcr{dimension}} of \mc{V}. 

\item[\tcr{Definition:}] A basis \mc{\cB = \{\vb_1,\ldots, \vb_k\}} of \mc{V} is called an \emph{\tcr{orthogonal basis}} if we have 
\beq
\label{1}
\tcb{\vb_i\cdot\vb_j =0,}\quad \mbox{for all } \tcb{i\ne j.}
\eeq 
It is called an \tcr{\emph{orthonormal basis}} if it is an orthogonal basis which satisfies \mc{||\vb_i|| =1,} for all \mc{i}.

\item[\tcr{Note:}] If \mc{\vb_1,\ldots,\vb_k} is a set of \tcr{non-zero} vectors which satisfies \sref{1}, then they are linearly independent.
 
\item[\tcr{Theorem 9:}] Let  \tcb{$\vec{b}_{\scr1}, \ldots, \vec{b}_{\scr k}$} be an \tcr{orthogonal basis} of \mc{V \subset \R^n}. Then the \tcg{orthogonal projection} of \mc{\vec{y}\in \R^n}
\tcg{onto} \tcb{$V$} is given by:\vspace*{-15pt}
$$
\tcb{P_V(\vec{y}) = \sum_{\scr i=1}^{\scr k} 
\frac{\vec{y}\cdot\vec{b}_{\scr i}}{\vec{b_{\scr i}}\cdot\vec{b}_{\scr i}}\vec{b}_{\scr i}.}    
$$
 
\end{description}
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