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The basic construction, given a vector space ''V'' over a division ring ''K'', is to form the set of . If ''K'' is the Real Number s, and ''V'' has dimension ''n'', then the projective space P(''V'') - which we can talk about as the space of lines through the zero element 0 of ''V'' - carries a natural structure of a Compact smooth Manifold of dimension ''n'' − 1. It is also highly symmetric, since any linear Automorphism of ''V'' gives rise to a symmetry of P(''V''). These in the classical examples identify with 'perspectivity' and 'projectivity' transformations described geometrically, and account for the name. The group of these symmetries is the quotient of the General Linear Group of ''V'' by the subgroup of non-zero scalar multiples of the identity. The use of projective spaces makes quite rigorous the talk about a ' and others become part of a theory founded on Linear Algebra . The part of a projective space not 'at infinity' is called Affine Space ; but the symmetries of P(''V'') do not respect that division. Use of a basis of ''V'' allows, if required, the introduction of Homogeneous Co-ordinates for the handling of concrete calculations.
MORPHISMS Projective linear maps between two projective spaces over the same field, say, ''P''(''V'') and ''P''(''W''), have the form
where ''T'' is an element of ''L''(''V'',''W''), the space of Linear Map s between ''V'' and ''W'', v is an element of ''V'', and we consider the equivalence classes 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 . On the other hand if ''T'' is not injective, it will have a Null Space larger than {0}; in this case the meaning of the class of ''T''(''v'') is problematic if ''v'' is non-zero and in the null space. What to do in that case falls under Birational Geometry . Two linear maps ''S'' and ''T'' in ''L''(''V'',''W'') induce the same map between ''P''(''V'') and ''P''(''W'') Iff they differ by a scalar multiple of the identity, that is if ''T''=''kS'' for some ''k'' ≠ 0. Thus if one identifies the scalar multiples of the Identity Map with the underlying field, the set of Morphism s from ''P''(''V'') to ''P''(''W'') is simply ''P''(''L''(''V'',''W'')).
SEE ALSO
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