Editing Projective space (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.