projective basis
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 $1,0,1$, and $2$, and then use the fact that any other projective space is isomorphic^{} to the projective space $P\beta \x81\u2019(V)$ associated with a vector space^{} $V$, and then define the dimension to be the dimension of $V$, minus $1$. In this entry, we introduce a more natural way of defining dimensions, via the concept of a basis.
Throughout the discussion, $\mathrm{\pi \x9d\x90\x8f}$ is a projective space (as in any model satisfying the axioms of projective geometry).
Given a subset $S$ of $\mathrm{\pi \x9d\x90\x8f}$, the span of $S$, written $\beta \x9f\xa8S\beta \x9f\copyright $, is the smallest subspace of $\mathrm{\pi \x9d\x90\x8f}$ containing $S$. In other words, $\beta \x9f\xa8S\beta \x9f\copyright $ is the intersection^{} of all subspaces of $\mathrm{\pi \x9d\x90\x8f}$ containing $S$. Thus, if $S$ is itself a subspace of $\mathrm{\pi \x9d\x90\x8f}$, $\beta \x9f\xa8S\beta \x9f\copyright =S$. We also say that $S$ spans $\beta \x9f\xa8S\beta \x9f\copyright $.
One may think of $\beta \x9f\xa8\beta \x8b\x85\beta \x9f\copyright $ as an operation^{} on the powerset of $\mathrm{\pi \x9d\x90\x8f}$. It is easy to verify that this operation is a closure operator^{}. In addition^{}, $\beta \x9f\xa8\beta \x8b\x85\beta \x9f\copyright $ is algebraic, in the sense that any point in $\beta \x9f\xa8S\beta \x9f\copyright $ is in the span of a finite subset of $S$. In other words,
$$\beta \x9f\xa8S\beta \x9f\copyright =\{P\beta \x88\pounds P\beta \x88\x88\beta \x9f\xa8F\beta \x9f\copyright \beta \x81\u2019\text{\Beta for some finite\Beta}\beta \x81\u2019F\beta \x8a\x86S\}.$$ 
Another property of $\beta \x9f\xa8\beta \x8b\x85\beta \x9f\copyright $ is the exchange property: for any subspace $U$, if $P\beta \x88\x89U$, then for any point $Q$, $\beta \x9f\xa8U\beta \x88\u037a\{P\}\beta \x9f\copyright =\beta \x9f\xa8U\beta \x88\u037a\{Q\}\beta \x9f\copyright $ iff $Q\beta \x88\x88\beta \x9f\xa8U\beta \x88\u037a\{P\}\beta \x9f\copyright U$.
A subset $S$ of $\mathrm{\pi \x9d\x90\x8f}$ is said to be projectively independent, or simply independent, if, for any proper subset^{} ${S}^{\beta \x80\xb2}$ of $S$, the span of ${S}^{\beta \x80\xb2}$ is a proper subset of the span of $S$: $\beta \x9f\xa8{S}^{\beta \x80\xb2}\beta \x9f\copyright \beta \x8a\x82\beta \x9f\xa8S\beta \x9f\copyright $. This is the same as saying that $S$ is a minimal^{} spanning set for $\beta \x9f\xa8S\beta \x9f\copyright $, in the sense that no proper subset of $S$ spans $\beta \x9f\xa8S\beta \x9f\copyright $. Equivalently, $S$ is independent iff for any $x\beta \x88\x88S$, $\beta \x9f\xa8S\{x\}\beta \x9f\copyright \beta \x89\beta \x9f\xa8S\beta \x9f\copyright $.
$S$ is called a projective basis, or simply basis for $\mathrm{\pi \x9d\x90\x8f}$, if $S$ is independent and spans $\mathrm{\pi \x9d\x90\x8f}$.
All of the properties about spanning sets, independent sets^{}, and bases for vector spaces have their projective counterparts. We list some of them here:

1.
Every projective space has a basis.

2.
If ${S}_{1},{S}_{2}$ are independent, then $\beta \x9f\xa8{S}_{1}\beta \x88\copyright {S}_{2}\beta \x9f\copyright =\beta \x9f\xa8{S}_{1}\beta \x9f\copyright \beta \x88\copyright \beta \x9f\xa8{S}_{2}\beta \x9f\copyright $.

3.
If $S$ is independent and $P\beta \x88\x88\beta \x9f\xa8S\beta \x9f\copyright $, then there is $Q\beta \x88\x88S$ such that $(\{P\}\beta \x88\u037aS)\{Q\}$ spans $\beta \x9f\xa8S\beta \x9f\copyright $.

4.
Let $B$ be a basis for $\mathrm{\pi \x9d\x90\x8f}$. If $S$ spans $\mathrm{\pi \x9d\x90\x8f}$, then $B\beta \x89\u20acS$. If $S$ is independent, then $S\beta \x89\u20acB$. As a result, all bases for $\mathrm{\pi \x9d\x90\x8f}$ have the same cardinality.

5.
Every independent subset in $\mathrm{\pi \x9d\x90\x8f}$ may be extended to a basis for $\mathrm{\pi \x9d\x90\x8f}$.

6.
Every spanning set for $\mathrm{\pi \x9d\x90\x8f}$ may be reduced to a basis for $\mathrm{\pi \x9d\x90\x8f}$.
In light of items 1 and 4 above, we may define the dimension of $\mathrm{\pi \x9d\x90\x8f}$ to be the cardinality of its basis.
One of the main result on dimension is the dimension formula^{}: if $U,V$ are subspaces of $\mathrm{\pi \x9d\x90\x8f}$, then
$$dim\beta \x81\u2018(U)+dim\beta \x81\u2018(V)=dim\beta \x81\u2018(U\beta \x88\u037aV)+dim\beta \x81\u2018(U\beta \x88\copyright V),$$ 
which is the counterpart of the same formula for vector subspaces of a vector space (see this entry (http://planetmath.org/DimensionFormulaeForVectorSpaces)).
References
 1 A. Beutelspacher, U. Rosenbaum Projective Geometry, From Foundations to Applications, Cambridge University Press (2000)
Title  projective basis 
Canonical name  ProjectiveBasis 
Date of creation  20130322 19:14:38 
Last modified on  20130322 19:14:38 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  10 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 05B35 
Classification  msc 06C10 
Classification  msc 51A05 
Synonym  independent 
Defines  span 
Defines  projective independence 
Defines  projectively independent 
Defines  basis 