We study the rank of complex sparse matrices in which the supports of
different columns have small intersections. The rank of these matrices, called
design matrices, was the focus of a recent work by Barak et. al. (BDWY11) in
which they were used to answer questions regarding point configurations. In
this work we derive near-optimal rank bounds for these matrices and use them to
obtain asymptotically tight bounds in many of the geometric applications. As a
consequence of our improved analysis, we also obtain a new, linear algebraic,
proof of Kelly's theorem, which is the complex analog of the Sylvester-Gallai
theorem