projective basis

In the parent entry, we see how one may define dimensionPlanetmathPlanetmathPlanetmath of a projective space inductively, from its subspacesPlanetmathPlanetmathPlanetmath starting with a point, then a line, and working its way up. Another way to define dimension start with defining dimensions of the empty setMathworldPlanetmath, 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 isomorphicPlanetmathPlanetmath to the projective space P⁒(V) associated with a vector spaceMathworldPlanetmath 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 intersectionMathworldPlanetmath 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 βŸ¨β‹…βŸ© as an operationMathworldPlanetmath on the powerset of 𝐏. It is easy to verify that this operation is a closure operatorPlanetmathPlanetmath. In additionPlanetmathPlanetmath, βŸ¨β‹…βŸ© 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∣P∈⟨F⟩⁒ for some finite ⁒FβŠ†S}.

Another property of βŸ¨β‹…βŸ© is the exchange property: for any subspace U, if Pβˆ‰U, then for any point Q, ⟨Uβˆͺ{P}⟩=⟨Uβˆͺ{Q}⟩ iff Q∈⟨Uβˆͺ{P}⟩-U.

A subset S of 𝐏 is said to be projectively independent, or simply independent, if, for any proper subsetMathworldPlanetmathPlanetmath Sβ€² of S, the span of Sβ€² is a proper subset of the span of S: ⟨Sβ€²βŸ©βŠ‚βŸ¨S⟩. This is the same as saying that S is a minimalPlanetmathPlanetmath spanning set for ⟨S⟩, in the sense that no proper subset of S spans ⟨S⟩. Equivalently, S is independent iff for any x∈S, ⟨S-{x}βŸ©β‰ βŸ¨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 setsMathworldPlanetmath, and bases for vector spaces have their projective counterparts. We list some of them here:

  1. 1.

    Every projective space has a basis.

  2. 2.

    If S1,S2 are independent, then ⟨S1∩S2⟩=⟨S1⟩∩⟨S2⟩.

  3. 3.

    If S is independent and P∈⟨S⟩, then there is Q∈S such that ({P}βˆͺS)-{Q} spans ⟨S⟩.

  4. 4.

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

  5. 5.

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

  6. 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 formulaMathworldPlanetmathPlanetmath: if U,V are subspaces of 𝐏, then


which is the counterpart of the same formula for vector subspaces of a vector space (see this entry (


  • 1 A. Beutelspacher, U. Rosenbaum Projective Geometry, From Foundations to Applications, Cambridge University Press (2000)
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