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

1.
$v\ne 0$ and

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 nonzero vector is isotropic, or that $Q(V)=0$.
Similarly, an isotropic quadratic form is one which has a nontrivial 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: $\mathrm{ker}(Q)=0$; totally isotropic: $\mathrm{ker}(Q)=V$).
Examples.

•
Consider the quadratic form^{} $Q(x,y)={x}^{2}+{y}^{2}$ in the vector space^{} ${\mathbb{R}}^{2}$ over the reals. It is clearly anisotropic since there are no real numbers $a,b$ not both $0$, such that ${a}^{2}+{b}^{2}=0$.

•
However, the same form is isotropic in ${\u2102}^{2}$ over $\u2102$, since ${1}^{2}+{i}^{2}=0$; the complex numbers^{} are algebraically closed^{}.

•
Again, using the same form ${x}^{2}+{y}^{2}$, but in ${\mathbb{R}}^{3}$ over the reals , we see that it is isotropic since the $z$ term is missing, so that $Q(0,0,1)={0}^{2}+{0}^{2}=0$.

•
If we restrict $Q$ to the subspace^{} consisting of the $z$axis ($x=y=0$) and call it ${Q}_{z}$, then ${Q}_{z}$ is totally isotropic, and the $z$axis is a totally isotropic subspace.

•
The quadratic form $Q(x,y)={x}^{2}{y}^{2}$ is clearly isotropic in any vector space over any field. In general, this is true if the coefficients of a diagonal quadratic form^{} $Q$ consist of $1,1,0$ ($0$ is optional) and nothing else.
Title  isotropic quadratic space 
Canonical name  IsotropicQuadraticSpace 
Date of creation  20130322 15:41:57 
Last modified on  20130322 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 