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affine hull

{{short description|Smallest affine subspace that contains a subset}}

{{One source|date=June 2022}}

In mathematics, the affine hull or affine span of a set S in Euclidean space Rn is the smallest affine set containing S,{{harvnb|Roman|2008|loc=p. 430 §16}} 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 of S is what \operatorname{span} S would be if the origin was moved to S.

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

:\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 R3 is the entire space R3.

Properties

For any subsets S, T \subseteq X

  • \operatorname{aff}(\operatorname{aff} S) = \operatorname{aff} S \subset \operatorname{span} S = \operatorname{span} \operatorname{aff} S.
  • \operatorname{aff} S is a closed set if X is finite dimensional.
  • \operatorname{aff}(S + T)=\operatorname{aff} S + \operatorname{aff} T.
  • S\subset \operatorname{aff} S.
  • If 0 \in \operatorname{aff} S then \operatorname{aff} S = \operatorname{span} S.
  • If s_0 \in \operatorname{aff} S then \operatorname{aff}(S) - s_0 = \operatorname{span}(S - s_0)= \operatorname{span}(S - S) is a linear subspace of X.
  • \operatorname{aff}(S - S) = \operatorname{span}(S - S) if S\ne\varnothing.
  • So, \operatorname{aff}(S - S) is always a vector subspace of X if S\ne\varnothing.
  • If S is convex then \operatorname{aff}(S - S) = \displaystyle\bigcup_{\lambda > 0} \lambda (S - S)
  • For every s_0 \in \operatorname{aff} S, \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 \operatorname{cone}(S - S) is the smallest cone containing S - S (here, a set C \subseteq X is a cone if r c \in C for all c \in C and all non-negative r \geq 0).
  • Hence \operatorname{cone}(S - S)= \operatorname{span}(S - S) is always a linear subspace of X parallel to \operatorname{aff} S if S\ne\varnothing.
  • Note: \operatorname{aff} S = s_0 + \operatorname{span}(S - s_0) says that if we translate S so that it contains the origin, take its span, and translate it back, we get \operatorname{aff} S. Moreover, \operatorname{aff} S or s_0 + \operatorname{span}(S - s_0) is what \operatorname{span} S would be if the origin was at s_0.

Related sets

  • If instead of an affine combination one uses a convex combination, that is, one requires in the formula above that all \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 \operatorname{cone} S.
  • If however one puts no restrictions at all on the numbers \alpha_i, instead of an affine combination one has a linear combination, and the resulting set is the linear span \operatorname{span} S of S, which contains the affine hull of S.

References

{{reflist}}

Sources

  • R.J. Webster, Convexity, Oxford University Press, 1994. {{ISBN|0-19-853147-8}}.
  • {{citation | last=Roman | first=Stephen

| title=Advanced Linear Algebra | edition=Third | series=Graduate Texts in Mathematics | publisher = Springer | date=2008| pages= | isbn=978-0-387-72828-5 |author-link=Steven Roman}}

Category:Affine geometry

Category:Closure operators