Affine hull

Affine hull is a universal term of the mathematical theory of affine spaces. Closely related is the concept of linear hull. This is called the affine hull and connecting space, especially when the subset is a union of two or more affine subspaces themselves.

Definition and properties

Definition

Be the belonging to a vector space affine space and a subset of. Then the affine hull of the smallest affine subspace of containing the whole amount.

Construction

With the notation from the definition is chosen from an arbitrary point. He serves as the start point of the affine hull. Then the linear hull is formed to the amount of the compound vectors. is the set of all finite linear combinations of elements of, ie, the linear span of in the vector space, which belongs to the affine space. This part of the structure is described in more detail in the article Linear Case. Now is the affine hull of.

The affine hull of the empty set is the empty set.

Properties

The affine hull of any subset of an affine space

  • Is uniquely determined ( as a specific amount, not just up to isomorphism )
  • Is an affine space with a dimension between -1 ( empty set) and the dimension of the total space,
  • Contains the convex hull of the set, and is also the affine case, if a real affinity space.

The mapping that assigns to every subset of an affine space their linear span, is a closure operator.

In the set of affine subspaces of an affine space (including the empty set and the whole space ) can be the operation " constitutes the affine hull of the union of " two-digit shortcut introduce here is when are written for these affine hull, it is then also referred to as a connecting space of subspaces. The corresponding dual link is then the intersection of education. With these links then forms a distributive lattice, and even a Boolean algebra.

  • For the dimensions of the connection space and the cut of two affine subspaces there is a dimension formula, see Affine subspace.

Examples

  • The affine hull of any two different points in space whose connecting line.
  • The affine hull of three points in space is a straight line, if the three points lie on a common straight line, otherwise the plane lie on all three points.
  • The affine hull of a plane figure in space ( triangle, circle, etc. ) is the plane containing the figure.
  • The affine hull of the set of polynomials is the family of curves. This example makes it clear that the affine hull is usually not a vector space.
Mathematics
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