Spaces of Singular Matrices and Matroid Parity

Let V be a linear space of even dimension n over a field F of characteristic 0. A subspace W ⊂ ∧ 2 V is maximal singular ifrank (w) ≤ n − 1 for allw ∈ W and any W⫋W′ ⊂ ∧ 2V contains a nonsingular matrix. shown that if W ⊂ ∧ 2V is a maximal singular subspace which is generated by decomposable elements thendimW ≥ 3n2 − 3 and that this bound is sharp. The main tool in the proof is the Lovász Matroid Parity Theorem.