basisuavhengig

Basisuavhengig is a concept used in linear algebra and related fields to describe properties, quantities, or constructions that do not depend on the particular choice of basis for a vector space. The term combines basis with avhengig (independent) in Scandinavian languages, and in English literature the corresponding notion is often called basis-independent or basis-invariant.

Definition and scope: A property P of a vector space, a subspace, a linear map, or a set of vectors is basisuavhengig if P remains unchanged under a change of basis. For example, the dimension of a vector space, the dimension of a subspace, the rank of a matrix, and the kernel or image of a linear map are basisuavhengig. Similarly, the span of a set is a subspace defined by its elements rather than by any particular basis.

Examples and contrasts: Coordinates of a vector or the explicit matrix representation of a linear map are basis-dependent, because they change when the basis changes. In contrast, coordinate-free notions and constructions—such as dimension, linear independence (as a property of a set), and subspace structure—are basisuavhengig.

Context and usage: The idea is widely used to emphasize formulations that do not rely on a specific coordinate system, aligning with coordinate-free or covariant approaches in mathematics and physics.

See also: basis, basis-invariant, basis-dependent, linear algebra, coordinate-free.