Let \(V\) be a real vector space with a symmetric bilinear form, or a complex vector space with a Hermitian form, and write \(\inner{\wait}{\wait}\) for the form. A vector \(v\in V\) is a null vector if \(\inner v w = 0\) for every \(w\in V\); the null vectors form the nullspace of the form. The form is nondegenerate if its nullspace is \(\theset 0\), that is, if for every \(v \neq 0\) there is some \(v'\in V\) with \(\inner{v}{v'} \neq 0\). Otherwise it is degenerate.
Let \(V\) and \(\inner{\wait}{\wait}\) be as in the definition, with \(V\) finite-dimensional.
Fix a basis of \(V\) and let \(A\) be the matrix of the form in that basis. Then \(v\) is a null vector if and only if its coordinate vector \(Y\) satisfies \(AY = 0\), so the form is nondegenerate if and only if \(A\) is invertible.
For a subspace \(W \subseteq V\) with \(W^\perp \coloneqq \theset{v\in V \st \inner w v = 0 \text{ for all } w\in W}\), the restriction of the form to \(W\) is nondegenerate if and only if \(W \cap W^\perp = \theset 0\), if and only if \(V = W \oplus W^\perp\).
Nondegeneracy of the form on \(V\) and nondegeneracy of its restriction to a subspace are independent. On \(V = \RR^2\) with standard basis \(e_1, e_2\), the form \(\inner x y = x_1y_1 - x_2y_2\) is nondegenerate, but its restriction to \(W = \RR(e_1 + e_2)\) is zero, hence degenerate. The form \(\inner x y = x_1y_1\) on \(\RR^2\) is degenerate, with nullspace \(\RR e_2\), but its restriction to \(\RR e_1\) is nondegenerate.
See [@Art11]. The formulation over an arbitrary field, through the adjoint \(V \to \dualof{V}\), is D-O4WWN.