Affine hull

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,

${\displaystyle \operatorname {aff} (S)=\left\{\sum _{i=1}^{k}\alpha _{i}x_{i}\,{\Bigg |}\,k>0,\,x_{i}\in S,\,\alpha _{i}\in \mathbb {R} ,\,\sum _{i=1}^{k}\alpha _{i}=1\right\}.}$