In mathematics, the affine hull or affine span of a set ${\displaystyle S}$ in Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ is the smallest affine set containing ${\displaystyle S}$, or equivalently, the intersection of all affine sets containing ${\displaystyle S}$. Here, an affine set may be defined as the translation of a vector subspace.

The affine hull of ${\displaystyle S}$ is what ${\displaystyle \operatorname {span} S}$ would be if the origin was moved to ${\displaystyle S}$.

The affine hull aff(${\displaystyle S}$) of ${\displaystyle S}$ is the set of all affine combinations of elements of ${\displaystyle S}$, that is,

For any subsets ${\displaystyle S,T\subseteq X}$