Affine hull

In mathematics, the affine hull or affine span of a set $S$ in Euclidean space $\mathbb {R} ^{n}$ 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 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 $\mathbb {R} ^{3}$ is the entire space $\mathbb {R} ^{3}$.

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 $S\neq \varnothing$.
  • So, $\operatorname {aff} (S-S)$ is always a vector subspace of $X$ if $S\neq \varnothing$.
• 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 $rc\in C$ 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 $S\neq \varnothing$.
  • 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 $S$.

Sources

• 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