Affine Function

linear-algebra analysis

Definition

Affine Function

An affine function is a function consisting of a linear part plus a constant shift.

For vector spaces over a field $F$, a function $f:F^n\to F^m$ is affine if there exist a matrix $A\in F^{m\times n}$ and a vector $b\in F^m$ such that

$$f(x)=Ax+b.$$

The matrix $A$ gives the linear transformation, while $b$ translates the output. If $b=0$, the affine function is linear.

Interpretation

Affine functions preserve straight-line structure but may move the origin. They map affine combinations to affine combinations:

$$f((1-\lambda )x+\lambda y)=(1-\lambda )f(x)+\lambda f(y)\lambda \in [0,1].$$

In one-dimensional optimisation, an affine function is a line. In higher dimensions, scalar-valued affine functions define hyperplanes as their level sets.