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Affine hull - Wikipedia

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Smallest affine subspace that contains a subset

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

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

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

aff ⁡ ( S ) = { ∑ i = 1 k α i x i | k > 0 , x i ∈ S , α i ∈ R , ∑ i = 1 k α i = 1 } . {\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\}.}

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

• R.J. Webster, Convexity, Oxford University Press, 1994. ISBN 0-19-853147-8.
• Roman, Stephen (2008), Advanced Linear Algebra, Graduate Texts in Mathematics (Third ed.), Springer, ISBN 978-0-387-72828-5

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