Betti numbers and linear covers of points

We prove that for a finite set of points X in the projective n-space over any field, the Betti number ₍, ₍+₁ of the coordinate ring of X is non-zero if and only if X lies on the union of two planes whose sum of dimension is less than n. Our proof is direct and short, and the inductive step rests on a combinatorial statement that works over matroids.