Isometry of a space with a form

A linear bijection between spaces carrying compatible bilinear forms, Hermitian forms or quadratic forms is an isometry of a space with a form when it preserves the specified form. Preserving the polar form alone need not preserve a quadratic form in characteristic two. Subspaces may have degenerate restricted forms even when the ambient form is nonsingular.

A vector is anisotropic for a quadratic form if $Q(v)\ne 0$, and for a Hermitian form if $h(v,v)\ne 0$. The complementary zero-value condition is isotropy. For a symmetric bilinear form in odd characteristic of a field, this agrees with the associated quadratic form. A nondegenerate alternating bilinear form has all self-pairings zero, so nondegeneracy must not be confused with the existence of anisotropic vectors.

An isometry of a space with a form between subspaces of a finite-dimensional nonsingular classical space extends to an ambient isometry. This includes alternating bilinear forms in all characteristics, symmetric bilinear forms in characteristic different from two, Hermitian forms with nontrivial involution, and quadratic forms with nonsingular polar form. No nonsingularity of the restricted subspace is required. Bare nonalternating symmetric bilinear forms in characteristic two require additional hypotheses and must not be confused with quadratic spaces.