< [[nlab:quadratic form]] ## Definiton ## Given a [[commutative ring]] $R$ and a $R$-[[module]] $B$, a quadratic form is a function $q:B \to R$ such that the binary function $r:B \times B \to R$ pointwise defined as $$r(u, v) \coloneqq q(u + v) - (q(u) + q(v))$$ is a [[bilinear function]], and for $a:R$ and $v:B$, $p(a, v): q(a v) = a^2 q(v)$. ## See also ## * [[module]] * [[bilinear function]] * [[Clifford algebra]] * [[geometric algebra]]