{{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 <math>S</math> in Euclidean space <math>\mathbb{R}^n</math> is the smallest affine set containing <math>S</math>,<ref>{{harvnb|Roman|2008|loc=p. 430 §16}}</ref> or equivalently, the intersection of all affine sets containing <math>S</math>. Here, an ''affine set'' may be defined as the translation of a vector subspace.
The affine hull of <math>S</math> is what <math>\operatorname{span} S</math> would be if the origin was moved to <math>S</math>.
The affine hull aff(<math>S</math>) of <math>S</math> is the set of all affine combinations of elements of <math>S</math>, that is,
:<math>\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\}.</math>
==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 <math>\mathbb{R}^3</math> is the entire space <math>\mathbb{R}^3</math>.
==Properties== For any subsets <math>S, T \subseteq X</math>
* <math>\operatorname{aff}(\operatorname{aff} S) = \operatorname{aff} S \subset \operatorname{span} S = \operatorname{span} \operatorname{aff} S</math>. * <math>\operatorname{aff} S</math> is a closed set if <math>X</math> is finite dimensional. * <math>\operatorname{aff}(S + T)=\operatorname{aff} S + \operatorname{aff} T</math>. * <math>S\subset \operatorname{aff} S</math>. * If <math>0 \in \operatorname{aff} S</math> then <math>\operatorname{aff} S = \operatorname{span} S</math>. * If <math>s_0 \in \operatorname{aff} S</math> then <math>\operatorname{aff}(S) - s_0 = \operatorname{span}(S - s_0)= \operatorname{span}(S - S)</math> is a linear subspace of <math>X</math>. * <math>\operatorname{aff}(S - S) = \operatorname{span}(S - S)</math> if <math>S\ne\varnothing</math>. ** So, <math>\operatorname{aff}(S - S)</math> is always a vector subspace of <math>X</math> if <math>S\ne\varnothing</math>. * If <math>S</math> is convex then <math>\operatorname{aff}(S - S) = \displaystyle\bigcup_{\lambda > 0} \lambda (S - S)</math> * For every <math>s_0 \in \operatorname{aff} S</math>, <math>\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)</math> where <math>\operatorname{cone}(S - S)</math> is the smallest cone containing <math>S - S</math> (here, a set <math>C \subseteq X</math> is a '''cone''' if <math>r c \in C</math> for all <math>c \in C</math> and all non-negative <math>r \geq 0</math>). ** Hence <math>\operatorname{cone}(S - S)= \operatorname{span}(S - S)</math> is always a linear subspace of <math>X</math> parallel to <math>\operatorname{aff} S</math> if <math>S\ne\varnothing</math>. ** Note: <math>\operatorname{aff} S = s_0 + \operatorname{span}(S - s_0)</math> says that if we translate <math>S</math> so that it contains the origin, take its span, and translate it back, we get <math>\operatorname{aff} S</math>. Moreover, <math>\operatorname{aff} S</math> or <math> s_0 + \operatorname{span}(S - s_0)</math> is what <math>\operatorname{span} S</math> would be if the origin was at <math>s_0</math>.
==Related sets==
*If instead of an affine combination one uses a convex combination, that is, one requires in the formula above that all <math>\alpha_i</math> be non-negative, one obtains the convex hull of <math>S</math>, which cannot be larger than the affine hull of <math>S</math>, as more restrictions are involved. *The notion of conical combination gives rise to the notion of the conical hull <math>\operatorname{cone} S</math>. *If however one puts no restrictions at all on the numbers <math>\alpha_i</math>, instead of an affine combination one has a linear combination, and the resulting set is the linear span <math>\operatorname{span} S</math> of <math>S</math>, which contains the affine hull of <math>S</math>.
==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