quadratic form
A quadratic form is the multivariable generalization of the single expression x^2. Just as squaring scales quadratically (doubling x quadruples x^2), a quadratic form Q on a vector space scales by Q(av) = a^2 Q(v), and its formula is a homogeneous polynomial of degree two in the coordinates — only terms like x_i x_j appear, never a lone x_i or a constant. It is the right object for measuring squared lengths, energies, and curvatures.
Concretely, over a field K a quadratic form on V is a map Q : V → K with Q(av) = a^2 Q(v) for all scalars a, such that the map B(u, v) = Q(u + v) - Q(u) - Q(v) is bilinear. This B is symmetric and is called the associated (polar) bilinear form. In coordinates Q(x) = sum over i ≤ j of a_ij x_i x_j, equivalently x^T A x for a symmetric matrix A when char K is not 2, with A the half-sum recovering Q on the diagonal.
When the characteristic is not 2, quadratic forms and symmetric bilinear forms determine each other: Q(v) = B(v, v) recovers Q from B, and polarization recovers B from Q. In characteristic 2 this correspondence fails, and a quadratic form carries strictly more information than its polar form; one then defines quadratic forms by the two axioms above and keeps the polar form merely as an auxiliary invariant.
Q(x, y) = x^2 + xy + y^2 over Q is a quadratic form; its polar bilinear form is B((x,y),(x',y')) = 2xx' + xy' + x'y + 2yy', symmetric with matrix [2, 1; 1, 2]. Note Q(v) = (1/2) B(v, v).
A binary quadratic form and its polar form over Q.
A ‘quadratic space’ is a vector space equipped with a quadratic form. The classification of quadratic spaces up to isometry — by rank, discriminant, signature, Hasse invariant — is one of the central goals of the whole field.