Functions

Linear

A functionm, \(f: \mathbb{R}^n \rightarrow \mathbb{R}\), is linear if

\[\begin{split}f(\alpha x + \beta y) = \alpha f(x) + \beta f(y) \\ \forall x, y \in \mathbb{R}^n \text{ and }\alpha, \beta \in \mathbb{R}\end{split}\]

property: \(f\) is linear if and only if \(f(x) = a^Tx\) for some \(a\)

Affine

A function, \(f: \mathbb{R}^n \rightarrow \mathbb{R}\), is affine if

\[\begin{split}f(\alpha x + (1-\alpha)y) = \alpha f(x) + (1-\alpha)f(y) \\ \forall x, y \in \mathbb{R}^n \text{ and }\alpha \in \mathbb{R}\end{split}\]

property: \(f\) is linear if and only if \(f(x) = a^Tx + b\) for some \(a\), \(b\)