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