Editing Projective space (section)

=== Subspace ===
Let {{math|'''P'''(''V'')}} be a projective space, where {{mvar|V}} is a vector space over a field {{mvar|K}}, and 
<math display="block">p:V\to \mathbf P(V)</math>
be the ''canonical map'' that maps a nonzero vector {{mvar|v}} to its equivalence class, which is the [[vector line]] containing {{mvar|v}} with the zero vector removed.

Every [[linear subspace]] {{mvar|W}} of {{mvar|V}} is a union of lines. It follows that {{math|''p''(''W'')}} is a projective space, which can be identified with {{math|'''P'''(''W'')}}.

A ''projective subspace'' is thus a projective space that is obtained by restricting to a linear subspace the equivalence relation that defines {{math|'''P'''(''V'')}}.

If {{math|''p''(''v'')}} and {{math|''p''(''w'')}} are two different points of {{math|'''P'''(''V'')}}, the vectors {{mvar|v}} and {{mvar|w}} are [[linearly independent]]. It follows that:
* There is exactly one projective line that passes through two different points of {{math|'''P'''(''V'')}}, and 
* A subset of {{math|'''P'''(''V'')}} is a projective subspace if and only if, given any two different points, it contains the whole projective line passing through these points.

In [[synthetic geometry]], where projective lines are primitive objects, the first property is an axiom, and the second one is the definition of a projective subspace.