for every , the function is onto.
is non-degenerate in the sense that implies .
An affine space is an affine space associated with some vector space. Elements of are called points of . The dimension of an affine space is just the dimension of its associated vector space. The function is called a direction. We sometimes write for .
For example, itself is an affine space associated with , with given by .
Vector Spaces Induced by an Affine Space
An affine space is often called a “vector space without the origin”. In other words, singling out a point in an affine space gives us a vector space, in the following sense: fix a point in an affine space , define a vector space as follows:
vectors of are points of ,
define by , where is the point determined by . Because of property 2 above, is uniquely determined.
define by , where is the point determined by . Again, is unique.
Both and are well-defined, because is a bijection. When there is no confusion, we may drop the subscript . It is also easy to verify that , together with and , is indeed a left vector space over , with as the origin, written . Furthermore, is isomorphic to . Hence, for any two points ; there is nothing special about , and any point of can be used as the origin of a vector space.
Continue to assume that is a left vector space over a division ring , and an affine space associated with . An affine subspace of is the collection of points of that is mapped to a vector subspace of by the induced function for some point . In other words, is the inverse image of under the function :
If is restricted to , then is an affine space associated with , since , given that .
For example, if is considered as an affine space associated with with the map , then an affine subspace of is just a coset of a subspace of . In other words, , where is a subspace of and is a vector. It is evident that is uniquely determined by , and up to translation by a vector in . In other words, any two cosets of are affinely isomorphic.
In an affine space , an affine point, affine line, or affine plane is a , or dimensional affine subspace. Thus, an affine point is just the inverse image of the origin . The codimension of an affine subspace is the codimension of the associated vector subspace. An affine hyperplane is an affine subspace with codimension 1. When there is no confusion, we may drop the word “affine” in affine point, affine line, etc… Affine subspaces are also called flats.
Affine geometry is, generally speaking, the study the geometric properties of affine subspaces. In particular, it is the study of the incidence structure on affine subspaces. Operationally, we may define an affine geometry of a vector space to be the poset of all affine subspaces of , orderd by set theoretic inclusion. Points in are commonly written without the set theoretic brackets, so that means .
Next, we can define an incidence relation on so that iff or . Together with , becomes an incidence geometry. Two flats and are said to be parallel if they have the same associated subspace. As a result, two parallel flats are never incident unless they are equal. Also, given a point not incident with , we can always find a flat incident with and parallel to . If with , simply take . This makes an affine incidence geometry.
In addition, we define to be the smallest flat in that contains both and . By Zorn’s lemma, exists. Since is also unique, is well-defined. This turns into an upper semilattice. If is the associated subspace of and is the associated subspace of , then is the associated subspace of . The definition of can be extended to an arbitrary set of flats, so that is the smallest flat that contains all flats in . In fact, it is not hard to see that is complete semilattice.
However, since may be empty, is not a lattice in general via the “meet” () operation. But when , . So is a partially defined operator on . If one adjoins the empty set to , then becomes a lattice. is called the null subspace and its dimension is defined to be . One can show that is a geometric lattice.
Although , it is not special, since all points are treated equally; there is no notion of an origin in . The notion of a metric is also absent, since the underlying vector space is not assumed to have an inner product. In fact, perpendicularity is not defined in . In contrast, both notions are important in Euclidean geometry, where an inner product has been defined, so that is the unique vector with length.
Affine versus Projective
Affine geometry and projective geometry are intimately related. Given an affine geometry one can construct projective geometries. One easy way is to identify flats that are parallel to each other. Because the parallel relation is an equivalence relation, we can partition into equivalence classes. Since each equivalence class is represented by exactly one subspace of , so can be identified with . Of course, can also be viewed as a sub-poset of (simply by taking all the subspaces of in ). More generally, if we fix any point , and take all flats that are incident with , the resulting subset forms a modular complemented geometric lattice that is isomorphic to . In fact, has the structure of a projective geometry.
Another way to construct a projective geometry from an affine one is to adjoin extra elements to . Remember that itself is not a lattice, but simply adjoining to won’t give us a projective geometry either, because the resulting lattice is not modular (take two parallel lines and a point lying on ; then , while ). We start by taking a vector space such that is a subspace of of codimension 1 (This can be done by linear algebra). Our objective is to show that is embeddable in .
Let be a non-zero vector and look at the affine hyperplane . Each affine subspace of the form in has the form in , where is a subspace of and . Let the collection of all affine subspaces (affine subspaces of that are incident with ). There is an obvious one-to-one order preserving correspondence between and .
Next, every affine subspace is the intersection of and a subspace of such that and . Just take . Clearly . In addition, , or else gives us a contradiction. So . Finally, if , then , where , , and . So . This implies . But , . Therefore . This means that .
The above paragraph shows there is a one-to-one order preserving map from to . If we delete all subspaces of from , and call
then we actually get an order-preserving bijection between and .
|Date of creation||2013-03-22 15:58:20|
|Last modified on||2013-03-22 15:58:20|
|Last modified by||CWoo (3771)|
|Defines||associated linear subspace|
|Defines||dimension of an affine space|