Affine hull

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In mathematics, the affine hull or affine span of a set S in Euclidean space Rⁿ is the smallest affine set containing S,^[1] or equivalently, the intersection of all affine sets containing S. Here, an affine set may be defined as the translation of a vector subspace.

The affine hull aff(S) of S is the set of all affine combinations of elements of S, that is,

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