Exercise 4

Prove that the definition of incidence is independent of the choice of the representatives of $p$ and $L$. That is, if $p_{1}$, $p_{2}$, $p_{3}$, and $q_{1}$, $q_{2}$, $q_{3}$ are two triples of homogeneous coordinates for $p$, and $L_{1}$, $L_{2}$, $L_{3}$, and $M_{1}$, $M_{2}$, $M_{3}$ are two triples of homogeneous coordinates for $L$, prove that $p_{1}L_{1}+p_{2}L_{2}+p_{3}L_{3}=0$ if and only if $q_{1}M_{1}+q_{2}M_{2}+q_{3}M_{3}=0$.