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Incidence Matrices of Finite Quadratic Spaces

In this paper, we first prove the 2-rank of full incidence matrix of PG(n,q) with $q$ an odd prime power. Then by the quadratic form defined on PG(n,q), the points of it are classified as isotropic and anisotropic points. We divide the full incidence matrix into four sub-matrices. Then by using the software package Magma, we give a general conjecture for the 2-rank of sub-matrices of the full incidence matrix in PG(n,q) and prove it in the case of n=1, 2.