Null vector

This article does not cite any references or sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (June 2009)

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 or 0 or simply 0. A zero vector has no direction.

A different kind of vector, also called null vector or zero vector, arises in various generalizations of Euclidean space, as explained below.

Since the word null has a more general (and very different) meaning in computer programming, many programmers prefer the term zero vector to avoid confusion. For example, the statement if ( MyVector == Null ) would intuitively be interpreted as if MyVector is a null pointer by many programmers, as opposed to if MyVector is a null/zero vector.

Linear algebra

For a general vector space, the zero vector (or null 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.

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.

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. 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.