nondegenerate form
A nondegenerate form has no ‘dead’ directions. If a vector is orthogonal to absolutely everything in the space, a nondegenerate form forces it to be the zero vector — there is no nonzero direction the form is totally blind to. This is the condition that makes a form a faithful pairing, capable of identifying the space with its own dual.
Formally, a bilinear form B on V is nondegenerate (or nonsingular) if its radical is zero: the only v with B(v, w) = 0 for all w in V is v = 0. Equivalently the induced map V → V*, sending v to B(v, −), is injective, hence an isomorphism in finite dimensions. In matrix terms B is nondegenerate iff its Gram matrix is invertible, iff its determinant is nonzero, iff the rank equals dim V.
Nondegeneracy is the standing hypothesis for the deep theory: orthogonal complements then satisfy dim W + dim W-perp = dim V and (W-perp)-perp = W, Witt's theorem applies, and orthogonal groups act transitively on vectors of a fixed nonzero value. Be careful to distinguish nondegenerate from anisotropic: x^2 − y^2 is nondegenerate yet has isotropic vectors, so nondegeneracy permits self-orthogonal vectors while forbidding only vectors orthogonal to the entire space.
On R^2 the form with Gram matrix [0, 1; 1, 0] (i.e. B(x, y) = x_1 y_2 + x_2 y_1) is nondegenerate since det = −1 ≠ 0, even though both e_1 and e_2 are isotropic: B(e_1, e_1) = 0 yet B(e_1, e_2) = 1, so e_1 is not orthogonal to all of V.
A nondegenerate form whose basis vectors are isotropic.