Zero vector

For other uses, see Null (disambiguation).

In mathematics, a null vector is an element of a vector space that in some appropriate sense has zero magnitude.

In a vector space with a bilinear form, a vector that is self-orthogonal (i.e. on which the bilinear form is zero) is referred to as a null vector. In a seminormed vector space, it refers to a vector with zero seminorm. In contrast, the term zero vector refers to the unique additive identity of the vector space.

In contexts in which the only null vector is the zero vector (such as Euclidean vector space) or where there is no defined concept of magnitude, null vector may be used as a synonym for zero vector.

Linear algebra

For a general vector space, the zero vector is the 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 is called kernel or null space.

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

The zero vector is both parallel and perpendicular to every vector.

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
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