Editing Quadratic form (section)

=== Quadratic space ===
{{see also|Bilinear form#Associated quadratic form}}
Given an {{math|''n''}}-dimensional [[vector space]] {{math|''V''}} over a field {{math|''K''}}, a ''quadratic form'' on {{math|''V''}} is a [[function (mathematics)|function]] {{math|''Q'' : ''V'' → ''K''}} that has the following property: for some basis, the function {{math|''q''}} that maps the coordinates of {{math|''v'' ∈ ''V''}} to {{math|''Q''(''v'')}} is a quadratic form. In particular, if {{math|1=''V'' = ''K''<sup>''n''</sup>}} with its [[standard basis]], one has 
<math display="block"> q(v_1,\ldots, v_n)= Q([v_1,\ldots,v_n])\quad \text{for} \quad [v_1,\ldots,v_n] \in K^n. </math>

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 {{math|''V''}}, although the quadratic form {{math|''q''}} depends on the choice of the basis.

A finite-dimensional vector space with a quadratic form is called a '''quadratic space'''.

The map {{math|''Q''}} is a [[homogeneous function]] of degree 2, which means that it has the property that, for all {{math|''a''}} in {{math|''K''}} and {{math|''v''}} in {{math|''V''}}:
<math display="block"> Q(av) = a^2 Q(v). </math>

When the characteristic of {{math|''K''}} is not 2, the bilinear map {{math|''B'' : ''V'' × ''V'' → ''K''}} over {{math|''K''}} is defined:
<math display="block"> B(v,w)= \tfrac{1}{2}(Q(v+w)-Q(v)-Q(w)).</math>
This bilinear form {{math|''B''}} is symmetric. That is, {{math|1=''B''(''x'', ''y'') = ''B''(''y'', ''x'')}} for all {{math|''x''}}, {{math|''y''}} in {{math|''V''}}, and it determines {{math|''Q''}}: {{math|1=''Q''(''x'') = ''B''(''x'', ''x'')}} for all {{math|''x''}} in {{math|''V''}}.

When the characteristic of {{math|''K''}} is 2, so that 2 is not a [[Unit (ring theory)|unit]], it is still possible to use a quadratic form to define a symmetric bilinear form {{math|1=''B''′(''x'', ''y'') = ''Q''(''x'' + ''y'') − ''Q''(''x'') − ''Q''(''y'')}}. However, {{math|''Q''(''x'')}} can no longer be recovered from this {{math|''B''′}} in the same way, since {{math|1=''B''′(''x'', ''x'') = 0}} for all {{math|''x''}} (and is thus alternating).<ref>This alternating form associated with a quadratic form in characteristic 2 is of interest related to the [[Arf invariant]] – {{citation|author=Irving Kaplansky | year=1974 | title=Linear Algebra and Geometry|page=27}}.</ref> Alternatively, there always exists a bilinear form {{math|''B''″}} (not in general either unique or symmetric) such that {{math|1=''B''″(''x'', ''x'') = ''Q''(''x'')}}.

The pair {{math|(''V'', ''Q'')}} consisting of a finite-dimensional vector space {{math|''V''}} over {{math|''K''}} and a quadratic map {{math|''Q''}} from {{math|''V''}} to {{math|''K''}} is called a '''quadratic space''', and {{math|''B''}} as defined here is the associated symmetric bilinear form of {{math|''Q''}}. The notion of a quadratic space is a coordinate-free version of the notion of quadratic form. Sometimes, {{math|''Q''}} is also called a quadratic form.

{{anchor|isometry}}Two {{math|''n''}}-dimensional quadratic spaces {{math|(''V'', ''Q'')}} and {{math|(''V''′, ''Q''′)}} are '''isometric''' if there exists an invertible linear transformation {{math|''T'' : ''V'' → ''V''′}} ('''isometry''') such that
<math display="block"> Q(v) = Q'(Tv) \text{ for all } v\in V.</math>

The isometry classes of {{math|''n''}}-dimensional quadratic spaces over {{math|''K''}} correspond to the equivalence classes of {{math|''n''}}-ary quadratic forms over {{math|''K''}}.