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\begin{definition}[Affine Function]
We say a function $A:\mathbb{R}^{m}\rightarrow \mathbb{R}^{n}$ is \textbf{%
affine} if there is a linear function $L:\mathbb{R}^{m}\rightarrow \mathbb{R}%
^{n}$ and a vector $b$ in $\mathbb{R}^{n}$ such that%
\begin{equation}
A(x)=L(x)+b
\end{equation}
\end{definition}

$\forall x$ in $\mathbb{R}^{m}$.

An \textit{affine} function is just a linear function plus a translation.
From our knowledge of linear functions, it follows that if $A:\mathbb{R}%
^{m}\rightarrow \mathbb{R}^{n}$ is $\mathit{affine}$, then there is an $%
n\times m$ matrix $M$ and a vector $b$ in $\mathbb{R}^{n}$ such that 
\begin{equation}
A(x)=Mx+b
\end{equation}

$\forall x$ in $\mathbb{R}^{m}$. In particular, if $f:\mathbb{R}\rightarrow 
\mathbb{R}$ is \textit{affine}, then there are real numbers $m$ and $b$ such
that%
\begin{equation}
f(x)=mx+b
\end{equation}

for all real numbers $x$.

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