the associated symmetric bilinear form of Q. The notion of a quadratic space is a coordinate-free version of the notion of quadratic form. Sometimes, Q is also called a quadratic form. Two n-dimensional quadratic spaces and are isometric if there exists an invertible linear transformation (isometry) such that The isometry classes of n-dimensional quadratic spaces over K correspond to the equivalence classes of n-ary quadratic forms over K. Generalization Let R be a commutative ring, M be an R-module, and be an R-bilinear form. A mapping is the associated quadratic form of b, and is the polar form of q. A quadratic form may be characterized in the following equivalent ways: There exists an R-bilinear form such that q(v) is the associated quadratic form. for all and , and the polar form of q is R-bilinear. Related concepts Two elements v and w of V are called orthogonal if . The kernel of a bilinear form B consists of the elements that are orthogonal to every element of V. Q is non-singular if the kernel of its associated bilinear form is {0}. If there exists a non-zero v in V such that , the quadratic form Q is isotropic, otherwise it is anisotropic. This terminology also applies to vectors and subspaces of a quadratic space. If the restriction of Q to a subspace U of V is identically zero, then U is totally singular. The orthogonal group of a non-singular quadratic form Q is the group of the linear automorphisms of V that preserve Q: that is, the group of isometries of into itself. If a quadratic space has a product so that A is an algebra over a field, and satisfies then it is a composition algebra. Equivalence of forms Every quadratic form q in n variables over a field of characteristic not equal to 2 is equivalent to a diagonal form Such a diagonal form is often denoted by Classification of all quadratic forms up to equivalence can thus be reduced to the case of diagonal forms. Geometric meaning Using Cartesian coordinates in three dimensions, let , and let be a symmetric 3-by-3 matrix. Then the geometric nature of the solution set of the equation depends on the eigenvalues of the matrix . If all eigenvalues of are non-zero, then the solution set is an ellipsoid or a hyperboloid. If all the eigenvalues are positive, then it is