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null vector wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 split algebras 2 examples 3 references toggle the table of contents null vector 9 languages чӑвашла español français bahasa indonesia 日本語 português română русский українська edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia vector on which a quadratic form is zero this article is about zeros of a quadratic form for the zero element in a vector space see zero vector for null vectors in minkowski space see minkowski space causal structure a null cone where q x y z x 2 y 2 z 2 displaystyle q x y z x 2 y 2 z 2 in mathematics given a vector space x with an associated quadratic form q written x q a null vector or isotropic vector is a non zero element x of x for which q x 0 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 x q which has a null vector is called a pseudo euclidean space the term isotropic vector v when q v 0 has been used in quadratic spaces 1 and anisotropic space for a quadratic space without null vectors a pseudo euclidean vector space may be decomposed non uniquely into orthogonal subspaces a and b x a 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 r 0 x a b q a q b r a a b b displaystyle bigcup _ r geq 0 x a b q a q b r a in a b in b the null cone is also the union of the isotropic lines through the origin split algebras edit a composition algebra with a null vector is a split algebra 2 in a composition algebra a the quadratic form is q x x x when x is a null vector then there is no multiplicative inverse for x and since x 0 a is not a division algebra in the cayley dickson construction the split algebras arise in the series bicomplex numbers biquaternions and bioctonions which uses the complex number field c displaystyle mathbb c as the foundation of this doubling construction due to l e dickson 1919 in particular these algebras have two imaginary units which commute so their product when squared yields 1 h i 2 h 2 i 2 1 1 1 displaystyle hi 2 h 2 i 2 1 1 1 then 1 h i 1 h i 1 h i 1 h i 1 h i 2 0 displaystyle 1 hi 1 hi 1 hi 1 hi 1 hi 2 0 so 1 hi is a null vector the real subalgebras split complex numbers split quaternions and split octonions with their null cones representing the light tracking into and out of 0 a suggest spacetime topology examples edit the light like vectors of minkowski space are null vectors the four linearly independent biquaternions l 1 hi n 1 hj m 1 hk and m 1 hk are null vectors and l n m m 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 3 in the verma module of a lie algebra there are null vectors references edit emil artin 1957 geometric algebra isotropic arthur a sagle ralph e walde 1973 introduction to lie groups and lie algebras page 197 academic press 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 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 vol 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 retrieved from https en wikipedia org w index php title null_vector oldid 1325743647 categories linear algebra quadratic forms hidden categories articles with short description short description is different from wikidata this page was last edited on 4 december 2025 at 22 34 utc page was rendered with parsoid text is available under the creative commons attribution sharealike 4 0 license additional terms may apply by using this site you agree to the terms of use and privacy policy wikipedia is a registered trademark of the wikimedia foundation inc a non profit organization privacy policy about wikipedia disclaimers contact wikipedia legal safety contacts code of 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