In the parent entry, we see how one may define dimension of a projective space inductively, from its subspaces starting with a point, then a line, and working its way up. Another way to define dimension start with defining dimensions of the empty set, a point, a line, and a plane to be , and , and then use the fact that any other projective space is isomorphic to the projective space associated with a vector space , and then define the dimension to be the dimension of , minus . In this entry, we introduce a more natural way of defining dimensions, via the concept of a basis.
Throughout the discussion, is a projective space (as in any model satisfying the axioms of projective geometry).
Given a subset of , the span of , written , is the smallest subspace of containing . In other words, is the intersection of all subspaces of containing . Thus, if is itself a subspace of , . We also say that spans .
One may think of as an operation on the powerset of . It is easy to verify that this operation is a closure operator. In addition, is algebraic, in the sense that any point in is in the span of a finite subset of . In other words,
Another property of is the exchange property: for any subspace , if , then for any point , iff .
A subset of is said to be projectively independent, or simply independent, if, for any proper subset of , the span of is a proper subset of the span of : . This is the same as saying that is a minimal spanning set for , in the sense that no proper subset of spans . Equivalently, is independent iff for any , .
is called a projective basis, or simply basis for , if is independent and spans .
All of the properties about spanning sets, independent sets, and bases for vector spaces have their projective counterparts. We list some of them here:
Every projective space has a basis.
If are independent, then .
If is independent and , then there is such that spans .
Let be a basis for . If spans , then . If is independent, then . As a result, all bases for have the same cardinality.
Every independent subset in may be extended to a basis for .
Every spanning set for may be reduced to a basis for .
In light of items 1 and 4 above, we may define the dimension of to be the cardinality of its basis.
One of the main result on dimension is the dimension formula: if are subspaces of , then
which is the counterpart of the same formula for vector subspaces of a vector space (see this entry (http://planetmath.org/DimensionFormulaeForVectorSpaces)).
- 1 A. Beutelspacher, U. Rosenbaum Projective Geometry, From Foundations to Applications, Cambridge University Press (2000)
|Date of creation||2013-03-22 19:14:38|
|Last modified on||2013-03-22 19:14:38|
|Last modified by||CWoo (3771)|