In mathematics, given a

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but ca ...

''X'' with an associated

quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong t ...

''q'', written , a null vector or isotropic vector is a non-zero element ''x'' of ''X'' for which . In the theory of

real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...

bilinear form In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called ''scalars''). In other words, a bilinear form is a function that is linear ...

definite quadratic form In linguistics, definiteness is a semantic feature of noun phrases, distinguishing between referents or senses that are identifiable in a given context (definite noun phrases) and those which are not (indefinite noun phrases). The prototypical ...

s and

isotropic quadratic form In mathematics, a quadratic form over a field ''F'' is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise the quadratic form is anisotropic. More precisely, if ''q'' is a quadratic form on a vector sp ...

s 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 In mathematics and theoretical physics, a pseudo-Euclidean space is a finite- dimensional real -space together with a non- degenerate quadratic form . Such a quadratic form can, given a suitable choice of basis , be applied to a vector , giving q( ...

. 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:

\bigcup_ \.

The null cone is also the union of the

isotropic line In the geometry of quadratic forms, an isotropic line or null line is a line for which the quadratic form applied to the displacement vector between any pair of its points is zero. An isotropic line occurs only with an isotropic quadratic form, an ...

s through the origin.

Examples

The

light-like In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...

vectors of

Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...

are null vectors. The four

linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts ar ...

biquaternion In abstract algebra, the biquaternions are the numbers , where , and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions c ...

s , , , and are null vectors and can serve as a

basis Basis may refer to: Finance and accounting *Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates *Basis trading, a trading strategy consisting of ...

for the subspace used to represent

spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...

. Null vectors are also used in the

Newman–Penrose formalism The Newman–Penrose (NP) formalism The original paper by Newman and Penrose, which introduces the formalism, and uses it to derive example results.Ezra T Newman, Roger Penrose. ''Errata: An Approach to Gravitational Radiation by a Method of Sp ...

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 Project Euclid is a collaborative partnership between Cornell University Library and Duke University Press which seeks to advance scholarly communication in theoretical and applied mathematics and statistics through partnerships with independent an ...

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

division algebra In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible. Definitions Formally, we start with a non-zero algebra ''D'' over a fiel ...

. In the

Verma module Verma modules, named after Daya-Nand Verma, are objects in the representation theory of Lie algebras, a branch of mathematics. Verma modules can be used in the classification of irreducible representations of a complex semisimple Lie algebra. Spe ...

of a

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...

there are null vectors.

References

* * * {{cite book , last = Neville , first = E. H. (Eric Harold) , author-link =Eric Harold Neville , title =Prolegomena to Analytical Geometry in Anisotropic Euclidean Space of Three Dimensions , publisher = Cambridge University Press , date = 1922 , pag
204
url =https://archive.org/details/prolegomenatoana00nevi Linear algebra Quadratic forms