• 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.

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

• ${\displaystyle \operatorname {aff} (\operatorname {aff} S)=\operatorname {aff} S}$ 
• ${\displaystyle \operatorname {aff} S}$  is a closed set if ${\displaystyle X}$  is finite dimensional.
• ${\displaystyle \operatorname {aff} (S+T)=\operatorname {aff} S+\operatorname {aff} T}$ 
• If ${\displaystyle 0\in S}$  then ${\displaystyle \operatorname {aff} S=\operatorname {span} S}$ .
• If ${\displaystyle s_{0}\in S}$  then ${\displaystyle \operatorname {aff} (S)-s_{0}=\operatorname {span} (S-s_{0})}$  is a linear subspace of ${\displaystyle X}$ .
• ${\displaystyle \operatorname {aff} (S-S)=\operatorname {span} (S-S)}$ .
  • So in particular, ${\displaystyle \operatorname {aff} (S-S)}$  is always a vector subspace of ${\displaystyle X}$ .
• If ${\displaystyle S}$  is convex then ${\displaystyle \operatorname {aff} (S-S)=\displaystyle \bigcup _{\lambda >0}\lambda (S-S)}$ 
• For every ${\displaystyle s_{0}\in S}$ , ${\displaystyle \operatorname {aff} S=s_{0}+\operatorname {cone} (S-S)}$  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 ${\displaystyle \operatorname {cone} (S-S)}$  is always a linear subspace of ${\displaystyle X}$  parallel to ${\displaystyle \operatorname {aff} S}$ .

• 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 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
• 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 of S, which contains the affine hull of 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