$V$ be a f.d, vector space over complex field , $P_1,...,P_n$ projection linear operators such that $P_1+...+P_n=I$ , then $P_iP_j=O,\forall i\ne j$?
Show that $P_i$ and $\sum_i P_i$ being idempotent implies $P_i P_j=\delta_{ij}$