Affine hull - Citizendia

In mathematics, the affine hull of a set S in Euclidean space Rⁿ is the smallest affine set containing S, or equivalently, the intersection of all affine sets containing S. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Affine geometry is a form of Geometry featuring the unique parallel line property (see the parallel postulate) but where the notion of angle is undefined and lengths In Mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently Here, an affine set may be defined as the translation of a vector subspace. In Euclidean geometry, a translation is moving every point a constant distance in a specified direction The concept of a linear subspace (or vector subspace) is important in Linear algebra and related fields of Mathematics.

The affine hull aff(S) of S is the set of all affine combinations of elements of S, that is,

$\operatorname{aff} (S)=\left\{\sum_{i=1}^k \alpha_i x_i \Bigg | x_i\in S, \, \alpha_i\in \mathbb{R}, \, \sum_{i=1}^k \alpha_i=1, k=1, 2, \dots\right\}.$

1 Examples
2 Properties
3 Related sets
4 References

Examples

The affine hull of a set of two different points is the line through them. In Mathematics, an affine combination of vectors x 1. x n is vector \sum_{i=1}^{n}{\alpha_{i} \cdot
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.

Properties

$aff(aff(S)) = aff(S)$
$aff(S)$ is a closed set

Related sets

If instead of an affine combination one uses a convex combination, that is one requires in the formula above that all $α i$ be non-negative, one obtains the convex hull of S, which must be smaller than the affine hull of S as more restrictions are involved. In Topology and related branches of Mathematics, a closed set is a set whose complement is open. A convex combination is a Linear combination of points (which can be vectors scalars, or more generally points in an Affine space) In Mathematics, the convex hull or convex envelope for a set of points X in a Real Vector space V is the minimal Convex

If however one puts no restrictions at all on the numbers $α i$ , instead of an affine combination one has a linear combination, and the resulting set is the linear span of S, which is bigger than the affine hull of S. In Mathematics, linear combinations are a concept central to Linear algebra and related fields of mathematics In the mathematical subfield of Linear algebra, the linear span, also called the linear hull, of a set of vectors in a Vector

References

R. J. Webster, Convexity, Oxford University Press, 1994. ISBN 0-19-853147-8.

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Contents

Examples

Properties

Related sets

References