\begin{definition}[Definition:Quadratic Form]
Let $\mathbb K$ be a field of characteristic $\Char {\mathbb K} \ne 2$.
Let $V$ be a vector space over $\mathbb K$.
A '''quadratic form''' on $V$ is a mapping $q : V \mapsto \mathbb K$ such that:
:$\forall v \in V : \forall \kappa \in \mathbb K : \map q {\kappa v} = \kappa^2 \map q v$
:$b: V \times V \to \mathbb K: \tuple {v, w} \mapsto \map q {v + w} - \map q v - \map q w$ is a bilinear form
\end{definition}