Null vector

{{#invoke:Hatnote|hatnote}} In linear algebra, the null vector or zero vector is the vector (0, 0, …, 0) in Euclidean space, all of whose components are zero. It is usually written with an arrow head above or below it : ${\vec {0}}$ or 0 or simply 0. A zero vector has arbitrary direction, but is orthogonal (i.e. perpendicular, normal) to all other vectors with the same number of components.

In vector spaces with an inner product for which the requirement of positive-definiteness has been dropped, a vector that has zero length is referred to as a null vector. The term zero vector is then still reserved for the additive identity of the vector spaces.

Linear algebra

For a general vector space, the zero vector is the uniquely determined vector that is the identity element for vector addition.

The zero vector is unique; if a and b are zero vectors, then a = a + b = b.

The zero vector is a special case of the zero tensor. It is the result of scalar multiplication by the scalar 0 (here meaning the additive identity of the underlying field, not necessarily the real number 0).

The preimage of the zero vector under a linear transformation f is called kernel or null space.

A zero space is a linear space whose only element is a zero vector.

The zero vector is, by itself, linearly dependent, and so any set of vectors which includes it is also linearly dependent.

In a normed vector space there is only one vector of norm equal to 0. This is just the zero vector. In vector algebra its coordinates are ( 0,0 ) and its unit vector is n

Seminormed vector spaces

In a seminormed vector space there might be more than one vector of norm equal to 0. These vectors are often called null vectors.

Examples

The light-like vectors of Minkowski space are null vectors. In general, the coordinate representation of a null vector in Minkowski space contains non-zero values.

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

Sources

• Linear Algebra (4th Edition), S. Lipcshutz, M. Lipson, Schaum’s Outlines, McGraw Hill (USA), 2009, ISBN 978-0-07-154352-1
• Applied Abstract Algebra, K.H. Kim, F.W. Roush, Ellis Horwood, John Wiley & Sons, 1983, (student) 0-85312-612-7 (library) ISBN 0-85312-563-5
• Vector Analysis (2nd Edition), M.R. Spiegel, S. Lipcshutz, D. Spellman, Schaum’s Outlines, McGraw Hill (USA), 2009, ISBN 978-0-07-161545-7
• Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010, ISBN 978-0-521-86153-3