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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Perpendicular Vectors and Othogonal Subspaces">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Fri Mar 28 12:47:55 1997}
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\section{Perpendicular Vectors and Orthogonal Subspaces\label%
{Perpendicular Vectors and Orthogonal Subspaces}}

\subsubsection{Length (Norm) of a Vector\label{length of a vector}}

If $x$ is a vector in $\UNICODE{0x211d} ^{n}$, then the \textbf{length} (or 
\textbf{norm}) of $x$ is defined by $\left| \left| x\right| \right| =\sqrt{%
x_{1}^{2}+x_{2}^{2}+...+x_{n}^{2}}$.

\subsubsection{Unit Vectors\label{unit vectors}}

Given any nonzero vector $x$, the vector $y=\frac{x}{\left| \left| x\right|
\right| }$ is a \textbf{unit vector }in the direction of $x$.

\subsubsection{Distance\label{distance}}

If $x=\left( x_{1},...,x_{m}\right) $ and $y=\left( y_{1},...,y_{m}\right) $
are two vectors in $\UNICODE{0x211d} ^{m}$, then the distance between $x$
and $y$ is defined to be $\left| \left| x-y\right| \right| =\sqrt{\left(
x_{1}-y_{1}\right) ^{2}+\left( x_{2}-y_{2}\right) ^{2}+...+\left(
x_{m}-y_{m}\right) ^{2}}$.

\subsubsection{Inner Product\label{inner product}}

The quantity $x^{T}y=\left[ 
\begin{array}{lll}
x_{1} & \cdots & x_{n}
\end{array}
\right] \left[ 
\begin{array}{l}
y_{1} \\ 
\vdots \\ 
y_{n}
\end{array}
\right]
=x_{1}y_{1}+x_{2}y_{2}+...+x_{n}y_{n}=\sum\limits_{i=1}^{n}x_{i}y_{i} $ is
called the \textbf{inner} (or \textbf{dot}) \textbf{product} of $x$ and $y$
in $\UNICODE{0x211d} ^{n}$.

\begin{itemize}
\item  $x$ and $y$ are \textbf{orthogonal\label{orthogonal vectors}} if
their inner product is zero.
\end{itemize}

\subsubsection{Orthogonal Subspaces\label{orthogonal subspaces}}

Two subspaces $V$ and $W$ of the vector space $\UNICODE{0x211d} ^{n}$ are 
\textbf{orthogonal} if every vector $v$ in $V$ is orthogonal to every vector 
$w$ in $W$; that is, $v^{T}w=0$ for all $v$ and $w$.

\subsubsection{Orthogonal Complement\label{orthogonal complement}}

Given a subspace $V$ of $\UNICODE{0x211d} ^{n}$, the space of all vectors
orthogonal to $V$ is called the \textbf{orthogonal complement} of $V$, and
is denoted $V^{\bot }$.

\subsubsection{Fundamental Theorem of Linear Algebra, Part II\label%
{fundamental theorem of linear algebra, part 2}}

\begin{itemize}
\item  The \hyperref{nullspace}{}{}{MA02_05.tex#subspaces} is the orthogonal
complement of the \hyperref{row space}{}{}{MA02_05.tex#subspaces} in $%
\UNICODE{0x211d} ^{n}$.

\item  The \hyperref{left nullspace}{}{}{MA02_05.tex#subspaces} is the
orthogonal complement of the \hyperref{column space}{}{}{%
MA02_05.tex#subspaces} in $\UNICODE{0x211d} ^{n}$.
\end{itemize}

\subsection{\ Related Scientific Notebook Topics}

{\small \hyperref{Computing inner product with matrix multiplication}{}{}{%
../../Help/DM8-2.tex#Matrix product}}

{\small \hyperref{Computing vector inner product}{}{}{%
../../Help/DM9-1.tex#Dot product}}

{\small \hyperref{Norm}{}{}{../../Help/DM8-5.tex#Norm}}

{\small \hyperref{Orthogonal matrices and orthogonality test}{}{}{%
../../Help/DM8-8.tex#Orthogonal matrix}\label{Orthonormal bases}}

{\small \hyperref{Orthogonal matrices and singular value decomposition}{}{}{%
../../Help/DM8-8.tex#SVD}}

{\small \hyperref{Orthogonal matrices, orthonormal bases, and QR
factorization}{}{}{../../Help/DM8-7.tex#QR factorization}}

\noindent \hrulefill

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\noindent \hrulefill

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