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)).