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In mathematics, given a vector space ''X'' with an associated quadratic form ''q'', written , a null vector or isotropic vector is an 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 there exists a nonzero null vector. Where such a vector exists, is called a pseudo-Euclidean space. A pseudo-Euclidean space may be decomposed (non-uniquely) into 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: : ==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.〔Patrick Dolan (1968) (A Singularity-free solution of the Maxwell-Einstein Equations ), Communications in Mathematical Physics 9(2):161–8, especially 166, link from Project Euclid〕 In the Verma module of a Lie algebra there are null vectors. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Null vector」の詳細全文を読む スポンサード リンク
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