EXAMPLES FROM LINEAR ALGEBRA AND PROJECTIVE GEOMETRY

As an introduction to the concepts of combinatorial geometry and matroid, we wish to emphasize those features of the theory that have given it a unifying role in other branches of mathematics, that have permitted it to be fruitfully applied in disparate domains of science, and that continue to arouse broader interest in the subject. It is our intention in this chapter to clarify the basic concepts by showing how they appear and are interrelated in a list of significant examples. This will give the reader a general orientation with respect to the basic concepts, prior to their axiomatic treatment in Chapter 2. Most of these examples will be dealt with in full detail in subsequent chapters.

The concept of combinatorial geometry arose from work in projective geometry and linear algebra. The focus of this work was to understand the basic properties of two relations:

1. (1) the incidence between points, lines, planes, and so on (which in general we call flats) in geometries and geometric configurations

2. (2) the linear dependence of sets of vectors.

The task to characterize (axiomatize) these relations of incidence and linear dependence seems in retrospect both urgent and feasible in the light of far-reaching applications, both to new geometries and to more general algebraic and combinatorial structures.

Research in combinatorial geometry has been concentrated on

1. (1) synthetic (combinatorial) methods, involving only incidence relations between flats, and the fundamental operations of projection and intersection

2. (2) intrinsic properties of configurations, internal properties that configurations possess independent of the way in which they may be represented or constructed within some conventional space.