CHAPTER 1: INTRODUCTIO N
1.1. Geometric, Linear and Projective Transformations
A geometric transformation g o f V ont o V
x
isabijectio n g:V+ V
x
whic h has the
following property fo r all subsets X o f V: X i s a subspace of V i f and only if gX i s a sub-
space of V
x
.
It is clear that a composition of geometric transformations i s geometric, and that the in-
verse of a geometric transformation i s also geometric. I f g: V * V
x
i s a geometric transform
tion, the n g preserve s inclusion, join, meet, Jordan-Holder chains, among subspaces. S o we
have the following proposition .
1.1.1. If g is a geometric transformation of V onto V x, then
g(un W)=gungw,g(u+ W)=gu+gw,
dimF gU = dimF U,
£0=0, gV=V
x
holds for all subspaces U and W of V.
By the projective space P(V) o f V w e mean the set of all subspaces of V. Thu s P(V)
consists of the elements of po w V whic h are subspaces of V; P{V) i s a partially ordered set,
the order relation being provided by set inclusion in V; an y two elements U an d W o f P(V)
have a join and a meet, namely the subspaces U + W an d £/ n W, s o that P(V) i s a lattice;
P(V) ha s an absolutely largest element V, an d an absolutely smallest element 0 ; t o each ele-
ment U o f P{V) w e attach the number dim
F
£/; eac h U i n P(V) ha s a Jordan-Holder chai
0 C C U, an d all such Jordan-Holder chains have length 1 + dim
F
U. Define
/»'(F)= {UeP(V)\dim FU=i}
and call P 1(F),i2(K),iy,-1(J0, th e set of lines, planes, hyperplanes, of V.
A projectivity n o f V ont o V
x
isabijectio n TT:P(V) » P(V X) whic h has the follow
ing property fo r all U, W in P(V): UCW i f and only if nUCnW.
It is clear that a composition o f projectivities is a projectivity, and that the inverse of a
projectivity i s also a projectivity. I f n:P(V)-P(V l) i s a projectivity o f V ont o V
l9
the n
n preserve s order, join, meet, Jordan-Holder chains, among the elements of P(V) an d P(V X).
So we have the following proposition .
1.1.2. / /
TT:P(V)+ P{V X)
is a projectivity of V onto V
v
then
2
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