Definition:Quadratic Form (Linear Algebra)

Definition

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:

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

Also see

• Definition:Bilinear Form (Linear Algebra)
• Definition:Associated Bilinear Form

• Results about quadratic forms in the context of linear algebra can be found here.

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