- Affine hull
In
mathematics , the affine hull of aset "S" inEuclidean space R"n" is the smallestaffine set containing "S", or equivalently, the intersection of all affine sets containing "S". Here, an "affine set" may be defined as the translation of avector subspace .The affine hull aff("S") of "S" is the set of all
affine combination s of elements of "S", that is,:operatorname{aff} (S)=left{sum_{i=1}^k alpha_i x_i Bigg | x_iin S, , alpha_iin mathbb{R}, , sum_{i=1}^k alpha_i=1, k=1, 2, dots ight}.
Examples
*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 R"3" is the entire space R"3".Properties
* mathrm{aff}(mathrm{aff}(S)) = mathrm{aff}(S)
* mathrm{aff}(S) is aclosed set 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 theconvex hull of "S", which must not be larger than the affine hull of "S" as more restrictions are involved.
*The notion ofconical combination gives rise to the notion of theconical hull
*If however one puts no restrictions at all on the numbers alpha_i, instead of an affine combination one has alinear combination , and the resulting set is thelinear span of "S", which is bigger than the affine hull of "S".References
* R.J. Webster, "Convexity", Oxford University Press, 1994. ISBN 0-19-853147-8.
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