Index
We can define any finite geometry by matrices! These matrices are referred to as incidence matrices. We will create these matrices by first enumerating the points and lines of our geometry, identifying the i th point with the i th row and the j th line with the j th column of the incidence matrix. Secondly, for a point i and line j, if point i lies on point j, then we put a one in the ij th position. But if point i does not lie on line j, then we put a zero in the ij th position.
Let's look at an example of a near-linear space: The incidence matrix of the following geometry
The great thing is that every matrix that defines a near-linear space will have certain properties. By axiom 1 of near-linear spaces, we know that any line is incident with at least two points. It follows then that any column of an incidence matrix for a near-linear geometry must have at least 2 ones. Similarly by axiom 2 we know that two distinct points are on at most one line. So given any pair of points we see that there is at most one column whose entries are nonzero in the rows corresponding to the pair. Thus, by looking at a given matrix we are able to tell if the figure to which it corresponds satisfies the axioms of a near linear space!
|
PROBLEM 3: (a) What properties should a matrix have if it is the incidence matrix for a linear geometry?
(b) Can the following matrix belong to a linear geometry?
Solutions
|
