nullvektor

The nullvektor, commonly called the zero vector, is the additive identity of a vector space. It is the unique element that, when added to any vector, leaves that vector unchanged: for every vector v in a vector space V, v plus the nullvektor equals v, and the same in the opposite order.

In a vector space over a field, the zero vector is defined by these properties and is unique: if w satisfies v + w = v for all v, then w must be the zero vector. In a coordinate representation, the nullvektor has all components zero. For example, in Euclidean space R^n it is (0, 0, ..., 0) and its norm is 0.

The nullvektor is closely tied to the geometric notion of the origin in coordinate spaces. In standard identification of a vector space with a coordinate system, the origin corresponds to the zero vector, though conceptually they belong to different structures (a point versus a vector).

Linear maps preserve the zero vector: for any linear transformation T: V → W, T(0) = 0. The concept extends to the null space (kernel) of a linear transformation, which is the set of all vectors mapped to zero. This set always contains the nullvektor and its dimension reflects the degrees of freedom in solving homogeneous systems.

In summary, the nullvektor serves as the foundational additive identity in vector spaces, with a unique, well-defined role across algebra, geometry, and linear analysis.