Null vector

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For null vectors as used in special relativity, see Minkowski space#Causal structure.

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.

For a general vector space, the 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.