Null Vector
Get Null Vector essential facts below. View Videos or join the Null Vector discussion. Add Null Vector to your PopFlock.com topic list for future reference or share this resource on social media.
Null Vector
A null cone where

In mathematics, given a vector space X with an associated quadratic form q, written , a null vector or isotropic vector is a non-zero element x of X for which .

In the theory of real bilinear forms, definite quadratic forms and isotropic quadratic forms are distinct. They are distinguished in that only for the latter does there exist a nonzero null vector.

A quadratic space which has a null vector is called a pseudo-Euclidean space.

A pseudo-Euclidean vector space may be decomposed (non-uniquely) into orthogonal subspaces A and B, , where q is positive-definite on A and negative-definite on B. The null cone, or isotropic cone, of X consists of the union of balanced spheres:

The null cone is also the union of the isotropic lines through the origin.

Examples

The light-like vectors of Minkowski space are null vectors.

The four linearly independent biquaternions , , , and are null vectors and can serve as a basis for the subspace used to represent spacetime. Null vectors are also used in the Newman-Penrose formalism approach to spacetime manifolds.[1]

A composition algebra splits when it has a null vector; otherwise it is a division algebra.

In the Verma module of a Lie algebra there are null vectors.

References

  • Dubrovin, B. A.; Fomenko, A. T.; Novikov, S. P. (1984). Modern Geometry: Methods and Applications. Translated by Burns, Robert G. Springer. p. 50. ISBN 0-387-90872-2.
  • Shaw, Ronald (1982). Linear Algebra and Group Representations. 1. Academic Press. p. 151. ISBN 0-12-639201-3.
  • Neville, E. H. (Eric Harold) (1922). Prolegomena to Analytical Geometry in Anisotropic Euclidean Space of Three Dimensions. Cambridge University Press. p. 204.

  This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Null_vector
 



 



 
Music Scenes