Definition:Quadratic Form

Definition
Let $\mathbb K$ be a field of characteristic $\operatorname{char}\mathbb K \neq 2$.

Let $V$ be a vector space over $\mathbb K$.

A quadratic form on $V$ is a mapping $q : V \mapsto \mathbb K$ such that:
 * $\forall v \in V : \forall \kappa \in \mathbb K : q(\kappa v) = \kappa^2 q(v)$
 * $b : V\times V \mapsto \mathbb K : (v,w) \mapsto q(v+w) - q(v) - q(w)$ is a bilinear form

Also defined as
A quadratic form is a homogeneous polynomial of degree $2$.

Example:
 * $x^2 + 2 x y - 3y^2 + 4 x z$

is a quadratic form in the variables $x$, $y$ and $z$.