1 Affine Function
In geometry and affine function (affine transformation, affine map) between two vector spaces consist of a linear transformation followed by a translation. In a geometric setting, these are precisely the functions that map straight lines to straight lines.
A function \(f:\mathbb{R}^m\to \mathbb{R}^n\) is linear if for any vectors \(\vec{x}\) and \(\vec{y}\) in \(\mathbb{R}^m\) \[ f(a \vec{x}+\vec{ y})=af(\vec{x})+f(\vec{y}). \] From results discussed in our notes on Linear Algebra we have that if \(f:\mathbb{R}^m\to \mathbb{R}^n\) is linear, then there exists vectors \(\vec{a}_{1}, \dots, \vec{a}_{n}\in \mathbb{R}^m\) such that for all \(\vec{ x}\in \mathbb{R}^m\) we have \[ f(\vec{x})=\begin{pmatrix}\vec{a}_{1}\cdot \vec{x} \\ \vdots \\ \vec{a}_{n} \vec{x}\end{pmatrix}. \] In matrix notation we can write a linear function as \[ f(\vec{x})=A_{1}x_{1}+\dots+A_{n}x_{n}, \] where \(A_{1}, \dots, A_{n}\) can be scalars, dense or sparse matrices.
A function \(g:\mathbb{R}^m\to \mathbb{R}^n\) is affine if there exists a linear function \(f:\mathbb{R}^m \to \mathbb{R}^n\) and a vector \(\vec{c}\in \mathbb{R}^n\) such that for all \(\vec{ x}\in \mathbb{R}^m\) we have \[ g(\vec{ x})=f(\vec{ x})+\vec{ c}. \]