Degenerate Bilinear Form

A bilinear form B: V × V → F on a vector space V over a field F that has a nontrivial kernel: there exists a nonzero v in V such that B(v,w)=0 for every w in V. Equivalently the linear map V → V* induced by v ↦ B(v,−) is not injective.

A bilinear form B: V×V→K on a vector space (or module) whose radical {v∈V | B(v,w)=0 for all w∈V} is nontrivial. Equivalently its Gram matrix (with respect to some basis) is singular, so B fails to induce an isomorphism V→V* or a nondegenerate pairing on V.