# isotropic quadratic space

A vector $v$ (an element of $V$) in a quadratic space $(V,Q)$ is isotropic if

1. 1.

$v\neq 0$ and

2. 2.

$Q(v)=0$.

Otherwise, it is called anisotropic. A quadratic space $(V,Q)$ is isotropic if it contains an isotropic vector. Otherwise, it is anisotropic. A quadratic space $(V,Q)$ is totally isotropic if every one of its non-zero vector is isotropic, or that $Q(V)=0$.

Similarly, an isotropic quadratic form is one which has a non-trivial kernel, or that there exists a vector $v$ such that $Q(v)=0$. The definitions for that of an anisotropic quadratic form and that of a totally isotropic quadratic form should now be clear from the above discussion (anisotropic: $\operatorname{ker}(Q)=0$; totally isotropic: $\operatorname{ker}(Q)=V$).

Examples.

 Title isotropic quadratic space Canonical name IsotropicQuadraticSpace Date of creation 2013-03-22 15:41:57 Last modified on 2013-03-22 15:41:57 Owner CWoo (3771) Last modified by CWoo (3771) Numerical id 10 Author CWoo (3771) Entry type Definition Classification msc 15A63 Classification msc 11E81 Related topic QuadraticMap2 Related topic QuadraticForm Defines isotropic vector Defines isotropic quadratic form Defines anisotropic vector Defines anisotropic quadratic form Defines anisotropic quadratic space Defines totally isotropic quadratic space Defines totally isotropic quadratic form