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 $P(V)$ 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, $𝐏$ is a projective space (as in any model satisfying the axioms of projective geometry).

Given a subset $S$ of $𝐏$, the span of $S$, written $⟨S⟩$, is the smallest subspace of $𝐏$ containing $S$. In other words, $⟨S⟩$ is the intersection of all subspaces of $𝐏$ containing $S$. Thus, if $S$ is itself a subspace of $𝐏$, $⟨S⟩=S$. We also say that $S$ spans $⟨S⟩$.

One may think of $⟨\cdot ⟩$ as an operation on the powerset of $𝐏$. It is easy to verify that this operation is a closure operator. In addition, $⟨\cdot ⟩$ is algebraic, in the sense that any point in $⟨S⟩$ is in the span of a finite subset of $S$. In other words,

$$⟨S⟩=\{P\mid P\in ⟨F⟩\text{ for some finite }F\subseteq S\}.$$

Another property of $⟨\cdot ⟩$ is the exchange property: for any subspace $U$, if $P\notin U$, then for any point $Q$, $⟨U\cup \{P\}⟩=⟨U\cup \{Q\}⟩$ iff $Q\in ⟨U\cup \{P\}⟩-U$.

A subset $S$ of $𝐏$ is said to be projectively independent, or simply independent, if, for any proper subset $S^{\prime }$ of $S$, the span of $S^{\prime }$ is a proper subset of the span of $S$: $⟨S^{\prime }⟩\subset ⟨S⟩$. This is the same as saying that $S$ is a minimal spanning set for $⟨S⟩$, in the sense that no proper subset of $S$ spans $⟨S⟩$. Equivalently, $S$ is independent iff for any $x\in S$, $⟨S-\{x\}⟩\ne ⟨S⟩$.

$S$ is called a projective basis, or simply basis for $𝐏$, if $S$ is independent and spans $𝐏$.

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 $⟨S_1\cap S_2⟩=⟨S_1⟩\cap ⟨S_2⟩$.

3. If $S$ is independent and $P\in ⟨S⟩$, then there is $Q\in S$ such that $(\{P\}\cup S)-\{Q\}$ spans $⟨S⟩$.

4. Let $B$ be a basis for $𝐏$. If $S$ spans $𝐏$, then $|B|\le |S|$. If $S$ is independent, then $|S|\le |B|$. As a result, all bases for $𝐏$ have the same cardinality.

5. Every independent subset in $𝐏$ may be extended to a basis for $𝐏$.

6. Every spanning set for $𝐏$ may be reduced to a basis for $𝐏$.

In light of items 1 and 4 above, we may define the dimension of $𝐏$ to be the cardinality of its basis.

One of the main result on dimension is the dimension formula: if $U,V$ are subspaces of $𝐏$, then

$$dim(U)+dim(V)=dim(U\cup V)+dim(U\cap V),$$

which is the counterpart of the same formula for vector subspaces of a vector space (see this entry (http://planetmath.org/DimensionFormulaeForVectorSpaces)).

Title	projective basis
Canonical name	ProjectiveBasis
Date of creation	2013-03-22 19:14:38
Last modified on	2013-03-22 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