Basisunit

Basisunit is a conceptual unit of measurement used to quantify the size of a vector’s coordinate representation with respect to a fixed basis in a finite-dimensional vector space. It is defined in terms of the coordinate vector [v]_B of a vector v with respect to a basis B = {b1, ..., bn}. If a norm on R^n is chosen (commonly the L2 norm), the basisunit of v is defined as ||[v]_B||, i.e., the length of the coordinate vector in that norm. The basisunit is thus dependent on the chosen basis and on the chosen norm; it does not in general equal the geometric length of v unless the basis is orthonormal with respect to the corresponding inner product.

In the standard Euclidean space R^n with the standard basis, the basisunit coincides with the usual vector norm. For example, in R^3, a vector v = (3, 4, 0) has basisunit 5 under the L2 norm. If a non-orthonormal basis is used, the same vector v can have a different basisunit, reflecting how its coordinates change under a basis transformation.

Applications of the concept are primarily educational and analytical. Basisunit helps illustrate the basis-dependence of coordinate magnitudes and provides a metric for comparing representations of the same vector across different bases. It can be used in discussions of basis change, sparsity in different representations, or in teaching to separate geometric length from coordinate size.

See also: coordinate vector, basis, norm, change of basis. Notes: basisunit is a theoretical construct used to discuss representation lengths; in practice one usually reports either the vector’s coordinates or its geometric length.