## Definition

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

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

A quadratic form on $V$ is a mapping $q : V \mapsto \mathbb K$ such that:

$\forall v \in V : \forall \kappa \in \mathbb K : q \left({\kappa v}\right) = \kappa^2 q \left({v}\right)$
$b : V \times V \to \mathbb K: \left({v, w}\right) \mapsto q \left({v + w}\right) - q \left({v}\right) - q \left({w}\right)$ is a bilinear form

## Also defined as

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

Example:

$x^2 + 2 x y - 3 y^2 + 4 x z$

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