attachment:AffineFunction.tex of AffineMotion

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  10 %TCIDATA{Created=Sunday, January 30, 2005 13:04:52}
  11 %TCIDATA{LastRevised=Sunday, January 30, 2005 13:25:14}
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  16 \newtheorem{theorem}{Theorem}
  17 \newtheorem{acknowledgement}[theorem]{Acknowledgement}
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  23 \newtheorem{condition}[theorem]{Condition}
  24 \newtheorem{conjecture}[theorem]{Conjecture}
  25 \newtheorem{corollary}[theorem]{Corollary}
  26 \newtheorem{criterion}[theorem]{Criterion}
  27 \newtheorem{definition}[theorem]{Definition}
  28 \newtheorem{example}[theorem]{Example}
  29 \newtheorem{exercise}[theorem]{Exercise}
  30 \newtheorem{lemma}[theorem]{Lemma}
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  41 \begin{document}
  42 
  43 
  44 \begin{definition}[Affine Function]
  45 We say a function $A:\mathbb{R}^{m}\rightarrow \mathbb{R}^{n}$ is \textbf{%
  46 affine} if there is a linear function $L:\mathbb{R}^{m}\rightarrow \mathbb{R}%
  47 ^{n}$ and a vector $b$ in $\mathbb{R}^{n}$ such that%
  48 \begin{equation}
  49 A(x)=L(x)+b
  50 \end{equation}
  51 \end{definition}
  52 
  53 $\forall x$ in $\mathbb{R}^{m}$.
  54 
  55 An \textit{affine} function is just a linear function plus a translation.
  56 From our knowledge of linear functions, it follows that if $A:\mathbb{R}%
  57 ^{m}\rightarrow \mathbb{R}^{n}$ is $\mathit{affine}$, then there is an $%
  58 n\times m$ matrix $M$ and a vector $b$ in $\mathbb{R}^{n}$ such that 
  59 \begin{equation}
  60 A(x)=Mx+b
  61 \end{equation}
  62 
  63 $\forall x$ in $\mathbb{R}^{m}$. In particular, if $f:\mathbb{R}\rightarrow 
  64 \mathbb{R}$ is \textit{affine}, then there are real numbers $m$ and $b$ such
  65 that%
  66 \begin{equation}
  67 f(x)=mx+b
  68 \end{equation}
  69 
  70 for all real numbers $x$.
  71 
  72 \end{document}