Note: this entry is being rewritten in response to the discussion ”http://planetmath.org/?op=getmsg&id=6329euclidean space has no origin” to make it more useful.
, the set of -tuplets of elements of can be made into a geometric space in several ways by imposing various geometric structures on it. In this entry, we shall explore four such structures.
0.1 as an affine space
0.2 as a Euclidean space
In order to make into a Euclidean space, we specify a distance function as follows:
This function is also known as a metric, but one needs to be careful with the term ”metric” because it is sometimes also used to refer to an inner product.
0.3 as vector space
0.4 as an inner product space
In order to make into an inner product space we specify, in addition to the addition map and the scalar product map, another map, namely the inner product map, , which is defined as follows:
1 Relations between structures
Some of these geometries impose more structure on our space than others. Affine geometry imposes the least structure and inner product space imposes the most. The situation may be summarized in the following diagram, where the arrows mean that we can get from one geometry to the next by imposing more structure:
The four geometric structures described above may be characterized in terms of the symmetry groups which preserve the structures in question. These groups happen to be Lie groups and are as follows:
The symmetry group is , the set of all homogeneous linear transformations.
The symmetry group is , the set of all homogeneous orthogonal transformations.
4 Abuses of language
Below is the old entry, which will be erased as soon as it has been supplanted by the new entry.
Affine space of dimension over a field is , the set of http://planetmath.org/node/6617-tuplets of elements of . The reason that the finite dimensional vector space has such a special name “affine space” is because it is acted upon by the set of affine transformations, which are transforms of the form
and hence is the natural setting for affine geometry. (For more information on affine geometry, please see section 1.1.2 of the http://planetmath.org/node/3824topic entry on geometry.) It is worth mentioning that the name “affine space” is used primarily in geometry and in commutative algebra. (In algebra, one does not concern oneself with affine transforms and simply uses the term “affine space” as synonym for “vector space”).
|Date of creation||2013-03-22 12:03:50|
|Last modified on||2013-03-22 12:03:50|
|Last modified by||rspuzio (6075)|