Editing Projective space (section)

== Related concepts ==

=== 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.

=== Span ===
Every [[set intersection|intersection]] of projective subspaces is a projective subspace. It follows that for every subset {{mvar|S}} of a projective space, there is a smallest projective subspace containing {{mvar|S}}, the intersection of all projective subspaces containing {{mvar|S}}. This projective subspace is called the ''projective span'' of {{mvar|S}}, and {{mvar|S}} is a spanning set for it.

A set {{mvar|S}} of points is ''projectively independent'' if its span is not the span of any proper subset of {{mvar|S}}. If {{mvar|S}} is a spanning set of a projective space {{mvar|P}}, then there is a subset of {{mvar|S}} that spans {{mvar|P}} and is projectively independent (this results from the similar theorem for vector spaces). If the dimension of {{mvar|P}} is {{mvar|n}}, such an independent spanning set has {{math|''n'' + 1}} elements.

Contrarily to the cases of [[vector space]]s and [[affine space]]s, an independent spanning set does not suffice for defining coordinates. One needs one more point, see next section.

=== Frame ===
{{main|Projective frame}}

A ''projective frame'' or ''projective basis'' is an ordered set of points in a projective space that allows defining coordinates.{{sfn|Berger|2009|loc=chapter 4.4. Projective bases}} More precisely, in an {{mvar|n}}-dimensional projective space, a projective frame is a tuple of {{math|''n'' + 2}} points such that any {{math|''n'' + 1}} of them are independent; that is, they are not contained in a [[hyperplane]].

If {{mvar|V}} is an {{math|(''n'' + 1)}}-dimensional vector space, and {{mvar|p}} is the canonical projection from {{mvar|V}} to {{math|'''P'''(''V'')}}, then {{math|(''p''(''e''<sub>0</sub>), ..., ''p''(''e''<sub>''n''+1</sub>))}} is a projective frame if and only if {{math|(''e''<sub>0</sub>, ..., ''e''<sub>''n''</sub>)}} is a basis of {{mvar|V}} and the coefficients of {{math|''e''<sub>''n''+1</sub>}} on this basis are all nonzero. By rescaling the first {{mvar|n}} vectors, any frame can be rewritten as {{math|(''p''(''e''′<sub>0</sub>), ..., p(''e''′<sub>''n''+1</sub>))}} such that {{math|1=''e''′<sub>''n''+1</sub> = ''e''′<sub>0</sub> + ... + ''e''′<sub>''n''</sub>}}; this representation is unique up to the multiplication of all {{math|''e''′<sub>''i''</sub>}} with a common nonzero factor.

The ''projective coordinates'' or ''homogeneous coordinates'' of a point {{math|''p''(''v'')}} on a frame {{math|(''p''(''e''<sub>0</sub>), ..., ''p''(''e''<sub>''n''+1</sub>))}} with {{math|1=''e''<sub>''n''+1</sub> = ''e''<sub>0</sub> + ... + ''e''<sub>''n''</sub>}} are the coordinates of {{mvar|v}} on the basis {{math|(''e''<sub>0</sub>, ..., ''e''<sub>''n''</sub>)}}. They are only defined up to scaling with a common nonzero factor.

The ''canonical frame'' of the projective space {{math|'''P'''{{sub|''n''}}(''K'')}} consists of images by {{mvar|p}} of the elements of the canonical basis of {{math|''K''{{sup|''n''+1}}}} (that is, the [[tuples]] with only one nonzero entry, equal to 1), and the image by {{mvar|p}} of their sum.

=== Projective geometry ===
{{excerpt|Projective geometry|templates=-General geometry}}

=== Projective transformation ===
{{excerpt|Homography}}