Nollavektori

Nollavektori, or the zero vector, is the additive identity in a vector space. In a vector space V over a field F, the zero vector is denoted 0_V (often simply 0) and satisfies v + 0_V = v for every vector v in V.

The zero vector is unique: there is exactly one element of V that acts as the additive identity. It also serves as the additive inverse anchor, since for every v in V there exists a vector −v such that v + (−v) = 0_V. The zero vector has norm zero, and in inner product spaces it is orthogonal to every vector. In any coordinate representation, the zero vector corresponds to all coordinates being zero.

Examples include the zero vector in R^n, which is (0, 0, ..., 0); the zero polynomial in a space of polynomials; and the zero matrix in a space of matrices under standard addition. More generally, any set that forms a vector space over F has a zero vector, and any linear combination that yields the zero vector plays a central role in defining subspaces and linear independence.

The zero vector is foundational in linear algebra. It must lie in every subspace, and it is used to define kernels of linear maps, as well as to test for linear independence. In summary, the nollavektori is the unique element that leaves other vectors unchanged under addition and serves as the additive identity across all vector spaces.