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projectivity (Definition)

Let $ PG(V)$ and $ PG(W)$ be projective geometries, with $ V,W$ vector spaces over a field $ K$. A function $ p$ from $ PG(V)$ to $ PG(W)$ is called a projective transformation, or simply a projectivity if

  1. $ p$ is a bijection, and
  2. $ p$ is order preserving.

A projective property is any geometric property, such as incidence, linearity, etc... that is preserved under a projectivity.

From the definition, we see that a projectivity $ p$ carries 0 to 0, $ V$ to $ W$. Furthermore, it carries points to points, lines to lines, planes to planes, etc.. In short, $ p$ preserves linearity. Because $ p$ is a bijection, $ p$ also preserves dimensions, that is $ \operatorname{dim}(S)=\operatorname{dim}(p(S))$, for any subspace $ S$ of $ V$. Other properties preserved by $ p$ are incidence: if $ S\cap T\neq \varnothing$, then $ p(S)\cap p(T)\neq \varnothing$; and cross ratios.

Every bijective semilinear transformation defines a projectiviity. To see this, let $ f:V\to W$ be a semilinear transformation. If $ S$ is a subspace of $ V$, then $ f(S)$ is a subspace of $ W$, as $ x,y\in f(S)$, then $ x+y=f(a)+f(b)=f(a+b)\in f(S)$, and $ \alpha x={\beta}^{\theta}x={\beta}^{\theta}f(a)=f(\beta a)\in f(S)$, where $ \theta$ is an automorphism of the common underlying field $ K$. Also, if $ S$ is a subspace of a subspace $ T$ of $ V$, then $ f(S)$ is a subspace of $ f(T)$. Now if we define $ f^*:PG(V)\to PG(W)$ by $ f^*(S)=f(S)$, it is easy to see that $ f^*$ is a projectivity.

Conversely, if $ V$ and $ W$ are of finite dimension greater than $ 2$, then a projectivity $ p:PG(V)\to PG(W)$ induces a semilinear transformation $ \hat{p}:V\to W$. This highly non-trivial fact is the (first) fundamental theorem of projective geometry.

If the semilinear transformation induced by the projectivity $ p$ is in fact a linear transformation, then $ p$ is a collineation: three distinct collinear points are mapped to three distinct collinear points.

Remark. The definition given in this entry is a generalization of the definition typically given for a projective transformation. In the more restictive definition, a projectivity $ p$ is defined merely as a bijection between two projective spaces that is induced by a linear isomorphism. More precisely, if $ P(V)$ and $ P(W)$ are projective spaces induced by the vector spaces $ V$ and $ W$, if $ L:V\to W$ is a bijective linear transformation, then $ p=P(L):P(V)\to P(W)$ defined by

$\displaystyle P(L)[v]=[Lv]$
is the corresponding projective transformation. $ [v]$ is the homogeneous coordinate representation of $ v$. In this definition, a projectiity is always a collineation. In the case where the vector spaces are finite dimensional with specified bases, $ p$ is expressible in terms of an invertible matrix ($ Lv=Av$ where $ A$ is an invertible matrix).



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"projectivity" is owned by CWoo. [ full author list (2) ]
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See Also: polarities and forms, sesquilinear forms over general fields, perspectivity, projective space

Also defines:  collineation, projective transformation, projective property

Attachments:
central collineations (Definition) by Algeboy
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Cross-references: matrix, invertible, terms, expressible, bases, finite dimensional, representation, homogeneous coordinate, linear isomorphism, projective spaces, collinear, linear transformation, induced, fundamental theorem of projective geometry, induces, finite, easy to see, subspace of a subspace, automorphism, semilinear transformation, bijective, subspace, dimensions, preserves, planes, lines, points, property, order, bijection, function, field, vector spaces, projective geometries
There are 10 references to this entry.

This is version 7 of projectivity, born on 2006-06-09, modified 2006-06-17.
Object id is 7982, canonical name is Projectivity.
Accessed 2663 times total.

Classification:
AMS MSC: 51A05 (Geometry :: Linear incidence geometry :: General theory and projective geometries)
 51A10 (Geometry :: Linear incidence geometry :: Homomorphism, automorphism and dualities)
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