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Projective geometryIn mathematics, a projective space is a fundamental construction from any vector space. It generalises the projective plane that may be constructed from a three-dimensional vector space, over any field. While the theory of projective planes has an aspect that belongs to combinatorics too, that is absent in the general case. Projective space is basic in algebraic geometry, through the rich field of projective geometry developed in the nineteenth century but also in the constructions of the modern theory (based on graded algebras). Projective spaces and their generalisation to flag manifolds also play a big part in topology, the theory of Lie groups and algebraic groups, and their representation theory.
The basic construction, given a vector space V over a field K, is to form the set of equivalence classes of non-zero vectors in V under the relation of scalar proportionality: we consider v to be proportional to w if v = cw with c in K non-zero. This idea goes back to mathematical descriptions of perspective. If the field K is the real numbers, 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 manifoldIn mathematics, a manifold ''M is a type of space, characterised in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. Therefor 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 groupAbstract algebra Algebra Linear algebra Lie groups In mathematics, the general linear group of degree n over a field F (such as R or C , written as GL ''n F , is the group of n ''n invertible matrices with entries from F with the group operation that of o 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 ' line at infinityIn geometry and topology, the line at infinity is a line which is added to the real (affine) plane in order to give closure to incidence properties of the resulting projective plane. The line at infinity is also called the ideal line . In projective geome' (where parallel lines meet), or a ' plane at infinityIn projective geometry, the plane at infinity is a projective plane which is added to the affine 3-space in order to give it closure of incidence properties. The result of the addition is the projective 3-space,. If the affine 3-space is real, , then the' for three dimensions: a translation of the latter can be made as part of the projective space associated to a four-dimensional real vector space. In that way geometrical ideas introduced by PonceletJean-Victor Poncelet ( July 1, 1788 December 22, 1867) was a mathematician and engineer who did much to revive projective geometry. Born to a poor family in Metz, France, Poncelet won a scholarship to the lycee and then the Ecole Polytechnique where he st and others become part of a theory founded on linear algebraLinear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. Vector spaces are a central theme in modern mathematics; thus, linear algebra is wi. The part of a projective space not 'at infinity' is called affine spaceIn mathematics, an affine space may be defined somewhat abstractly as a set on which a vector space acts transitively. Albeit somewhat jocular, the following characterization may be easier to understand: an affine space is what is left of a vector space a; 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-ordinatesIn mathematics, homogeneous co-ordinates introduced by August Ferdinand Mobius, make calculations possible in projective space just as Cartesian co-ordinates do in Euclidean space. The homogeneous co-ordinates of a point of projective space of dimension n for the handling of concrete calculations.
Use of vector spaces over the field of complex numbers gives rise to different manifolds, also used by geometers. There are good reasons for using them, in order to get a theory about intersections of algebraic varieties with predictable properties. In the theory of Alexander Grothendieck there are reasons for applying the construction outlined above rather to the dual space V*.
