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Affine hull

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Smallest affine subspace that contains a subset
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In mathematics, the affine hull or affine span of a set S in Euclidean space R is the smallest affine set containing S, 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,

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\}.}

Examples

  • The affine hull of the empty set is the empty set.
  • The affine hull of a singleton (a set made of one single element) is the singleton itself.
  • The affine hull of a set of two different points is the line through them.
  • The affine hull of a set of three points not on one line is the plane going through them.
  • The affine hull of a set of four points not in a plane in R is the entire space R.

Properties

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

  • aff ⁡ ( aff ⁡ S ) = aff ⁡ S {\displaystyle \operatorname {aff} (\operatorname {aff} S)=\operatorname {aff} S}
  • aff ⁡ S {\displaystyle \operatorname {aff} S} is a closed set if X {\displaystyle X} is finite dimensional.
  • aff ⁡ ( S + T ) = aff ⁡ S + aff ⁡ T {\displaystyle \operatorname {aff} (S+T)=\operatorname {aff} S+\operatorname {aff} T}
  • If 0 ∈ S {\displaystyle 0\in S} then aff ⁡ S = span ⁡ S {\displaystyle \operatorname {aff} S=\operatorname {span} S} .
  • If s 0 ∈ S {\displaystyle s_{0}\in S} then aff ⁡ ( S ) − s 0 = span ⁡ ( S − s 0 ) = span ⁡ ( S − S ) {\displaystyle \operatorname {aff} (S)-s_{0}=\operatorname {span} (S-s_{0})=\operatorname {span} (S-S)} is a linear subspace of X {\displaystyle X} .
  • aff ⁡ ( S − S ) = span ⁡ ( S − S ) {\displaystyle \operatorname {aff} (S-S)=\operatorname {span} (S-S)} if S ≠ ∅ {\displaystyle S\neq \varnothing } .
    • So in particular, aff ⁡ ( S − S ) {\displaystyle \operatorname {aff} (S-S)} is always a vector subspace of X {\displaystyle X} if S ≠ ∅ {\displaystyle S\neq \varnothing } .
  • If S {\displaystyle S} is convex then aff ⁡ ( S − S ) = ⋃ λ > 0 λ ( S − S ) {\displaystyle \operatorname {aff} (S-S)=\displaystyle \bigcup _{\lambda >0}\lambda (S-S)}
  • For every s 0 ∈ S {\displaystyle s_{0}\in S} , aff ⁡ S = s 0 + span ⁡ ( S − s 0 ) = s 0 + span ⁡ ( S − S ) = S + span ⁡ ( S − S ) = s 0 + cone ⁡ ( S − S ) {\displaystyle \operatorname {aff} S=s_{0}+\operatorname {span} (S-s_{0})=s_{0}+\operatorname {span} (S-S)=S+\operatorname {span} (S-S)=s_{0}+\operatorname {cone} (S-S)} where cone ⁡ ( S − S ) {\displaystyle \operatorname {cone} (S-S)} is the smallest cone containing S − S {\displaystyle S-S} (here, a set C ⊆ X {\displaystyle C\subseteq X} is a cone if r c ∈ C {\displaystyle rc\in C} for all c ∈ C {\displaystyle c\in C} and all non-negative r ≥ 0 {\displaystyle r\geq 0} ).
    • Hence cone ⁡ ( S − S ) = span ⁡ ( S − S ) {\displaystyle \operatorname {cone} (S-S)=\operatorname {span} (S-S)} is always a linear subspace of X {\displaystyle X} parallel to aff ⁡ S {\displaystyle \operatorname {aff} S} if S ≠ ∅ {\displaystyle S\neq \varnothing } .

Related sets

  • If instead of an affine combination one uses a convex combination, that is, one requires in the formula above that all α i {\displaystyle \alpha _{i}} be non-negative, one obtains the convex hull of S, which cannot be larger than the affine hull of S, as more restrictions are involved.
  • The notion of conical combination gives rise to the notion of the conical hull cone ⁡ S {\displaystyle \operatorname {cone} S} .
  • If however one puts no restrictions at all on the numbers α i {\displaystyle \alpha _{i}} , instead of an affine combination one has a linear combination, and the resulting set is the linear span span ⁡ S {\displaystyle \operatorname {span} S} of S, which contains the affine hull of S.

References

  1. Roman 2008, p. 430 §16

Sources

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