In this paper we prove that balanced characteristic functions of canonical cliques in a Paley graph of square order $P(q^2)$ span the $\frac{-1+q}{2}$-eigenspace of the graph. This is the first of two steps to a second proof of the analogue of Erdös-Ko-Rado theorem for Paley graphs of square order (the first proof was given by A. Blokhuis in 1984).