Editing Projective space (section)

== Morphisms ==
Injective [[linear map]]s {{math|''T'' ∈ ''L''(''V'', ''W'')}} between two vector spaces {{math|''V''}} and {{math|''W''}} over the same field&nbsp;{{math|''K''}} induce mappings of the corresponding projective spaces {{math|'''P'''(''V'') → '''P'''(''W'')}} via:
{{block indent | em = 1.5 | text = {{math|[''v''] → [''T''(''v'')]}},}}
where {{math|''v''}} is a non-zero element of {{math|''V''}} and [...] denotes the equivalence classes of a vector under the defining identification of the respective projective spaces.  Since members of the equivalence class differ by a scalar factor, and linear maps preserve scalar factors, this induced map is [[well-defined]]. (If {{math|''T''}} is not injective, it has a [[null space]] larger than {{math|{{mset|0}}}}; in this case the meaning of the class of {{math|''T''(''v'')}} is problematic if {{math|''v''}} is non-zero and in the null space. In this case one obtains a so-called [[rational map]], see also ''[[Birational geometry]]''.)

Two linear maps {{math|''S''}} and {{math|''T''}} in {{math|''L''(''V'', ''W'')}} induce the same map between {{math|'''P'''(''V'')}} and {{math|'''P'''(''W'')}} [[if and only if]] they differ by a scalar multiple, that is if {{math|1=''T'' = ''λS''}} for some {{math|''λ'' ≠ 0}}.  Thus if one identifies the scalar multiples of the [[identity function|identity map]] with the underlying field&nbsp;{{math|''K''}}, the set of {{math|''K''}}-linear [[morphism]]s from {{math|'''P'''(''V'')}} to {{math|'''P'''(''W'')}} is simply {{math|1='''P'''(''L''(''V'', ''W''))}}.

The [[automorphism]]s {{math|'''P'''(''V'') → '''P'''(''V'')}} can be described more concretely. (We deal only with automorphisms preserving the base field&nbsp;{{math|''K''}}). Using the notion of [[ample line bundle|sheaves generated by global sections]], it can be shown that any algebraic (not necessarily linear) automorphism must be linear, i.e., coming from a (linear) automorphism of the vector space {{math|''V''}}. The latter form the [[group (mathematics)|group]] [[general linear group|{{math|GL(''V'')}}]]. By identifying maps that differ by a scalar, one concludes that

{{block indent | em = 1.5 | text ={{math|1=Aut('''P'''(''V'')) = Aut(''V'') / ''K''<sup>×</sup> = GL(''V'') / ''K''<sup>×</sup> =: PGL(''V'')}},}}
the [[quotient group]] of {{math|GL(''V'')}} modulo the matrices that are scalar multiples of the identity. (These matrices form the [[center of a group|center]] of {{math|Aut(''V'')}}.) The groups {{math|PGL}} are called [[projective linear group]]s. The automorphisms of the complex projective line {{math|'''P'''<sup>1</sup>('''C''')}} are called [[Möbius transformation]]s.