Affine hull

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

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

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

${\displaystyle \operatorname {aff} (S)=\left\{\sum _{i=1}^{k}\alpha _{i}x_{i}\,{\Bigg |}\,k>0,\,x_{i}\in S,\,\alpha _{i}\in \mathbb {R} ,\,\sum _{i=1}^{k}\alpha _{i}=1\right\}.}$

• 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 ${\displaystyle \mathbb {R} ^{3}}$ is the entire space ${\displaystyle \mathbb {R} ^{3}}$.

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

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

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

• 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