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A projective space of dimension 1 is a projective line, and a projective space of dimension 2 is a projective plane. Projective spaces are widely used in geometry , allowing for simpler statements and simpler proofs.
In mathematics, real projective space, denoted or (), is the topological space of lines passing through the origin 0 in the real space +. It is a compact , smooth manifold of dimension n , and is a special case G r ( 1 , R n + 1 ) {\displaystyle \mathbf {Gr} (1,\mathbb {R} ^{n+1})} of a Grassmannian space.
The projective axioms may be supplemented by further axioms postulating limits on the dimension of the space. The minimum dimension is determined by the existence of an independent set of the required size. For the lowest dimensions, the relevant conditions may be stated in equivalent form as follows. A projective space is of: (L1) at least ...
The projective plane cannot be embedded (that is without intersection) in three-dimensional Euclidean space. The proof that the projective plane does not embed in three-dimensional Euclidean space goes like this: Assuming that it does embed, it would bound a compact region in three-dimensional Euclidean space by the generalized Jordan curve ...
Formally, a complex projective space is the space of complex lines through the origin of an (n+1)-dimensional complex vector space. The space is denoted variously as P(C n+1), P n (C) or CP n. When n = 1, the complex projective space CP 1 is the Riemann sphere, and when n = 2, CP 2 is the complex projective plane (see there for a more ...
A projective plane is a 2-dimensional projective space. Not all projective planes can be embedded in 3-dimensional projective spaces; such embeddability is a consequence of a property known as Desargues' theorem, not shared by all projective planes.
The group of symmetries of the projective space is the group of projectivized linear automorphisms + (). The choice of a morphism to a projective space j : X → P n {\displaystyle j:X\to \mathbf {P} ^{n}} modulo the action of this group is in fact equivalent to the choice of a globally generating n -dimensional linear system of divisors on a ...
The projective space PG(n, q) consists of all the positive (algebraic) dimensional vector subspaces of V. An alternate way to view the construction is to define the points of PG( n , q ) as the equivalence classes of the non-zero vectors of V under the equivalence relation whereby two vectors are equivalent if one is a scalar multiple of the other.