nondegenerate form

A bilinear form is nondegenerate if the only vector orthogonal to everything is the zero vector — its radical is trivial. Put positively: for every nonzero v there exists some w with B(v, w) != 0, so the form can always 'feel' a nonzero vector through some test partner. Inner products are the gold-standard example, since B(v, v) > 0 already detects every nonzero v.

The matrix test is immediate: B is nondegenerate iff its matrix is invertible, iff det(B) != 0, iff rank(B) = n. So nondegeneracy is to forms what invertibility is to operators — the condition that nothing is lost, nothing collapses to zero. Degenerate forms, by contrast, have a kernel direction they cannot see.

The deep reason nondegeneracy matters is that it sets up an isomorphism between V and its dual space V*. Each vector v gives a functional w -> B(v, w); nondegeneracy says this assignment v -> B(v, -) is injective, and in finite dimensions injective plus equal dimensions means it is an isomorphism V -> V*. So a nondegenerate form is exactly the extra structure that canonically identifies vectors with covectors.

This is the abstract heart of 'raising and lowering indices' in tensor calculus and relativity, and of the Riesz representation idea in inner-product spaces. Without a nondegenerate form, V and V* are merely the same size but not naturally identified; with one, they become two faces of a single object. Nondegeneracy is the price of admission to that identification.