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 \vec{0} or 0 or simply 0.

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

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

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