mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, given a

vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...

''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

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 linea ...

s, 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 In mathematics and theoretical physics, a pseudo-Euclidean space of signature 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 vect ...

. The term ''isotropic vector v'' when ''q''(''v'') = 0 has been used in quadratic spaces, 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'', , 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 lines through the origin.

Split algebras

A composition algebra with a null vector is a split algebra. 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 In mathematics, the Cayley–Dickson construction, sometimes also known as the Cayley–Dickson process or the Cayley–Dickson procedure produces a sequence of algebra over a field, algebras over the field (mathematics), field of real numbers, eac ...

, the split algebras arise in the series bicomplex numbers, biquaternions, and bioctonions, which uses the

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

field

\Complex

as the foundation of this doubling construction due to L. E. Dickson (1919). In particular, these algebras have two

imaginary unit The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...

s, which commute so their product, when squared, yields +1: :

(hi)^2 = h^2 i^2 = (-1)(-1) = +1 .

Then :

(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

The light-like vectors of

Minkowski space In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model. The model helps show how a ...

are null vectors. The four

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

biquaternions , , , and are null vectors and can serve as a basis for the subspace used to represent

spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...

. 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 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 a ...

In the Verma module 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 Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...

, date = 1922 , pag
204
url =https://archive.org/details/prolegomenatoana00nevi Linear algebra Quadratic forms