is a bijection, and
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 carries 0 to 0, to . Furthermore, it carries points to points, lines to lines, planes to planes, etc.. In short, preserves linearity. Because is a bijection, also preserves dimensions, that is , for any subspace of . In particular, . Other properties preserved by are incidence: if , then ; and cross ratios (http://planetmath.org/CrossRatio).
Every bijective semilinear transformation defines a projectiviity. To see this, let be a semilinear transformation. If is a subspace of , then is a subspace of , as , then , and , where is an automorphism of the common underlying field . Also, if is a subspace of a subspace of , then is a subspace of . Now if we define by , it is easy to see that is a projectivity.
Conversely, if and are of finite dimension greater than , then a projectivity induces a semilinear transformation . This highly non-trivial fact is the (first) fundamental theorem of projective geometry.
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 is defined merely as a bijection between two projective spaces that is induced by a linear isomorphism. More precisely, if and are projective spaces induced by the vector spaces and , if is a bijective linear transformation, then defined by
is the corresponding projective transformation. is the homogeneous coordinate representation of . In this definition, a projectiity is always a collineation. In the case where the vector spaces are finite dimensional with specified bases, is expressible in terms of an invertible matrix ( where is an invertible matrix).
|Date of creation||2013-03-22 15:58:00|
|Last modified on||2013-03-22 15:58:00|
|Last modified by||CWoo (3771)|