mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the affine hull or affine span of a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

''S'' in

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

R^''n'' is the smallest

affine set In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...

containing ''S'', or equivalently, the

intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their i ...

of all affine sets containing ''S''. Here, an ''affine set'' may be defined as the

translation Translation is the communication of the Meaning (linguistic), meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The ...

of a

vector subspace In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, li ...

. The affine hull aff(''S'') of ''S'' is the set of all

affine combination In mathematics, an affine combination of is a linear combination : \sum_^ = \alpha_ x_ + \alpha_ x_ + \cdots +\alpha_ x_, such that :\sum_^ =1. Here, can be elements ( vectors) of a vector space over a field , and the coefficients \alpha_ a ...

s of elements of ''S'', that is, :

\operatorname (S)=\left\.

Examples

*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 R^''3'' is the entire space R^''3''.

Properties

For any subsets

S, T \subseteq X

\operatorname(\operatorname S) = \operatorname S

\operatorname S

is a

closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a cl ...

X

is finite dimensional. *

\operatorname(S + T)=\operatorname S + \operatorname T

* If

0 \in S

then

\operatorname S = \operatorname S

. * If

s_0 \in S

then

\operatorname(S) - s_0 = \operatorname(S - s_0)

is a linear subspace of

X

. *

\operatorname(S - S) = \operatorname(S - S)

. ** So in particular,

\operatorname(S - S)

is always a vector subspace of

X

. * If

S

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope ...

then

\operatorname(S - S) = \displaystyle\bigcup_ \lambda (S - S)

* For every

s_0 \in S

\operatorname S = s_0 + \operatorname(S - S)

where

\operatorname(S - S)

is the smallest

cone A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines con ...

containing

S - S

(here, a set

C \subseteq X

is a cone if

r c \in C

for all

c \in C

and all non-negative

r \geq 0

). ** Hence

\operatorname(S - S)

is always a linear subspace of

X

parallel to

\operatorname S

Related sets

*If instead of an affine combination one uses a

convex combination In convex geometry and vector algebra, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1. In other word ...

, that is one requires in the formula above that all

\alpha_i

be non-negative, one obtains the

convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...

of ''S'', which cannot be larger than the affine hull of ''S'' as more restrictions are involved. *The notion of

conical combination Given a finite number of vectors x_1, x_2, \dots, x_n in a real vector space, a conical combination, conical sum, or weighted sum''Convex Analysis and Minimization Algorithms'' by Jean-Baptiste Hiriart-Urruty, Claude Lemaréchal, 1993, pp. 101, 102/ ...

gives rise to the notion of the

conical hull Given a finite number of vectors x_1, x_2, \dots, x_n in a real vector space, a conical combination, conical sum, or weighted sum''Convex Analysis and Minimization Algorithms'' by Jean-Baptiste Hiriart-Urruty, Claude Lemaréchal, 1993, pp. 101, 102/ ...

*If however one puts no restrictions at all on the numbers

\alpha_i

, instead of an affine combination one has a linear combination, and the resulting set is the

linear span In mathematics, the linear span (also called the linear hull or just span) of a set of vectors (from a vector space), denoted , pp. 29-30, §§ 2.5, 2.8 is defined as the set of all linear combinations of the vectors in . It can be characterized ...

of ''S'', which contains the affine hull of ''S''.

References

* R.J. Webster, ''Convexity'', Oxford University Press, 1994. {{ISBN, 0-19-853147-8. Affine geometry Closure operators