Editing Quadratic form (section)

=== Related concepts ===
{{see also|Isotropic quadratic form}}
Two elements {{math|''v''}} and {{math|''w''}} of {{math|''V''}} are called '''[[orthogonal]]''' if {{math|1=''B''(''v'', ''w'') = 0}}. The '''kernel''' of a bilinear form {{math|''B''}} consists of the elements that are orthogonal to every element of {{math|''V''}}. {{math|''Q''}} is '''non-singular''' if the kernel of its associated bilinear form is {{math|{{mset|0}}}}. If there exists a non-zero {{math|''v''}} in {{math|''V''}} such that {{math|1=''Q''(''v'') = 0}}, the quadratic form {{math|''Q''}} is '''[[Isotropic quadratic form|isotropic]]''', otherwise it is '''[[definite quadratic form|definite]]'''. This terminology also applies to vectors and subspaces of a quadratic space. If the restriction of {{math|''Q''}} to a subspace {{math|''U''}} of {{math|''V''}} is identically zero, then {{math|''U''}} is '''totally singular'''.

The orthogonal group of a non-singular quadratic form {{math|''Q''}} is the group of the linear automorphisms of {{math|''V''}} that preserve {{math|''Q''}}:  that is, the group of isometries of {{math|(''V'', ''Q'')}} into itself.

If a quadratic space {{math|(''A'', ''Q'')}} has a product so that {{math|''A''}} is an [[algebra over a field]], and satisfies
<math display="block">\forall x, y \isin A \quad Q(x y) = Q(x) Q(y) ,</math> then it is a [[composition algebra]].