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

Improved rank bounds for design matrices and a new proof of Kelly's theorem