isotropic quadratic space
A vector (an element of ) in a quadratic space is isotropic if
Otherwise, it is called anisotropic. A quadratic space is isotropic if it contains an isotropic vector. Otherwise, it is anisotropic. A quadratic space is totally isotropic if every one of its non-zero vector is isotropic, or that .
Similarly, an isotropic quadratic form is one which has a non-trivial kernel, or that there exists a vector such that . 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: ; totally isotropic: ).
Again, using the same form , but in over the reals , we see that it is isotropic since the term is missing, so that .
If we restrict to the subspace consisting of the -axis () and call it , then is totally isotropic, and the -axis is a totally isotropic subspace.
|Title||isotropic quadratic space|
|Date of creation||2013-03-22 15:41:57|
|Last modified on||2013-03-22 15:41:57|
|Last modified by||CWoo (3771)|
|Defines||isotropic quadratic form|
|Defines||anisotropic quadratic form|
|Defines||anisotropic quadratic space|
|Defines||totally isotropic quadratic space|
|Defines||totally isotropic quadratic form|