Because it is so exciting that we can relate these geometries to matrices, lets look at some of the properties of matrices with which we are already familiar. For example, lets examine the transpose of the matrix introduced above.

In the transpose, our points become lines and our lines become points. Try to draw the figure that corresponds to this matrix. You will see that in the context of near-linear spaces, this transpose makes no sense! Certainly, a point can lie on only one line, but when we take the transpose, this means that a line will lie on only one point (as is the case with line a and point l₄). By Axiom 1, this is impossible. So to alleviate this problem, we must eliminate our rows that have less than two ones. Thus, our new matrix is:

And its transpose is

This makes much more sense within the context of near-linear geometries, and we can draw the geometry which corresponds to this transposed matrix.

We will call this the dual geometry of our original geometry.

But let's back track to see how we can come up with the dual geometry without using the incidence matrix.

Let's suppose that our geometry S has point set P and line set L. We will then define the dual space, R, as consisting of a new set of points P' and a new set of lines L' such that P' refers simply to L (P' = L), the lines of S. Then we define L' to be the set defined by the points of S with at least two lines passing through them. Can you then see why the transpose of our incidence matrix without the rows containing less than 2 ones is the incidence matrix of the dual geometry?

Note that this definition of dual is a bit different from that given by Polster: Here we are taking the dual of a near linear geometry in such a way as to guarantee that we get another near linear geometry.

References: Batten, Polster.