of q may be replaced by the symmetric matrix with the same quadratic form, so it may be assumed from the outset that A is symmetric. Moreover, a symmetric matrix A is uniquely determined by the corresponding quadratic form. Under an equivalence C, the symmetric matrix A of φ and the symmetric matrix B of ψ are related as follows: The associated bilinear form of a quadratic form q is defined by Thus, bq is a symmetric bilinear form over K with matrix A. Conversely, any symmetric bilinear form b defines a quadratic form and these two processes are the inverses of each other. As a consequence, over a field of characteristic not equal to 2, the theories of symmetric bilinear forms and of quadratic forms in n variables are essentially the same. Quadratic space Given an n-dimensional vector space V over a field K, a quadratic form on V is a function that has the following property: for some basis, the function q that maps the coordinates of to is a quadratic form. In particular, if with its standard basis, one has The change of basis formulas show that the property of being a quadratic form does not depend on the choice of a specific basis in V, although the quadratic form q depends on the choice of the basis. A finite-dimensional vector space with a quadratic form is called a quadratic space. The map Q is a homogeneous function of degree 2, which means that it has the property that, for all a in K and v in V: When the characteristic of K is not 2, the bilinear map over K is defined: This bilinear form B is symmetric. That is, for all x, y in V, and it determines Q: for all x in V. When the characteristic of K is 2, so that 2 is not a unit, it is still possible to use a quadratic form to define a symmetric bilinear form . However, Q(x) can no longer be recovered from this B′ in the same way, since for all x (and is thus alternating). Alternatively, there always exists a bilinear form B″ (not in general either unique or symmetric) such that . The pair consisting of a finite-dimensional vector space V over K and a quadratic map Q from V to K is called a quadratic space, and B as defined here is