Affine shell

Affine envelope is a universal term from the mathematical theory of affine spaces . The concept of the linear envelope is closely related . The affine hull is also called a connecting space , especially when the subset itself is a union of two or more affine subspaces . ${\ displaystyle M}$ ${\ displaystyle M = U \ cup V}$

Definition and characteristics

definition

Let be the affine space belonging to a vector space and a subset of . Then the affine hull of is the smallest affine subspace of that contains the whole set . ${\ displaystyle A}$ ${\ displaystyle \ mathbb {K}}$ ${\ displaystyle M \ subseteq A}$ ${\ displaystyle A}$ ${\ displaystyle M}$ ${\ displaystyle A}$ ${\ displaystyle M}$

construction

Any point can be selected using the designations from the definition . It serves as the starting point of the affine envelope. Then the linear envelope is formed for the set of connection vectors . is the set of all finite linear combinations of elements from , i.e. the linear envelope of in the vector space that belongs to the affine space . This part of the construction is detailed in the article Linear Case described. Now the affine hull of . ${\ displaystyle M}$ ${\ displaystyle P_ {0}}$ ${\ displaystyle V (M) = \ lbrace {\ overrightarrow {PQ}} \ mid P, Q \ in M \ rbrace}$ ${\ displaystyle H}$ ${\ displaystyle H}$ ${\ displaystyle V (M)}$ ${\ displaystyle V (M)}$ ${\ displaystyle A}$ ${\ displaystyle P_ {0} + H}$ ${\ displaystyle M}$

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

properties

The affine hull of any subset of an affine space ${\ displaystyle M}$ ${\ displaystyle A}$

is clearly determined (as a concrete set, not just down 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 its affine hull if it is a real affine space. ${\ displaystyle M}$ ${\ displaystyle A}$

The mapping that assigns its linear envelope to each subset of an affine space is an envelope operator .

In the set of affine subspaces of an affine space (including the empty set and the total space) the operation “form the affine hull of the union” can be introduced as a two-digit combination, here, if are, is written for this affine hull, it is then also referred to as the connecting space of the subspaces. The dual link for this is then the formation of intersections. With these links it then forms an association . ${\ displaystyle T}$ ${\ displaystyle U, V \ in T}$ ${\ displaystyle U \ vee V}$ ${\ displaystyle T}$

There is a dimension formula for the dimensions of the connecting space and the intersection of two affine subspaces, see Affine subspace .

Examples

The affine hull of any two different points in space is the straight line connecting them .

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 on which all three points lie.

The affine shell of a plane figure in space (triangle, circle, etc.) is the plane that contains the figure.

The affine hull of the polynomial set is the family of curves . This example makes it clear that the affine hull is usually not a vector space. ${\ displaystyle \ lbrace 1, x ^ {2}, x ^ {3} \ rbrace \ subseteq \ mathbb {R} [x]}$ ${\ displaystyle \ lbrace 1 + a (x ^ {2} -1) + b (x ^ {3} -1): a, b \ in \ mathbb {R} \ rbrace}$

literature

Uwe Storch, Hartmut Wiebe: Lineare Algebra (= textbook of mathematics for mathematicians, computer scientists and physicists . Volume II ). 2nd, corrected edition. BI-Wissenschafts-Verlag, Mannheim 1990, ISBN 3-8274-0359-6 ( table of contents [accessed December 25, 2011]).

Web links

Rolf Waldi: Description of connecting spaces (PDF; 60 kB) in the script additions to geometry .