Basisindex

Basisindex is a term used to describe the label or position of a basis element within a given basis for a vector space or function space. In a vector space V over a field F, a basis B is a set of vectors {b_i | i ∈ I} that spans V. The index i from the index set I serves as the basisindex, identifying each specific basis vector b_i.

In practical terms, basisindices are used to express vectors by coordinates relative to the basis. For a vector v in V, there exist coefficients v_i such that v = sum_{i ∈ I} v_i b_i. The coordinate vector [v]_B collects these coefficients with components indexed by the basisindices, typically written as (v_i) arranged in a corresponding order.

Basisindices also play a central role in changing basis and representing linear maps. When switching from one basis to another, the index labeling of basis vectors must be tracked to construct the appropriate change-of-basis matrices and to interpret the resulting coordinate representations correctly. In finite-dimensional spaces, indices are often taken to be integers 1 through n, aligning with conventionally ordered bases; in infinite-dimensional spaces, the index set I may be countable or uncountable, depending on the basis.

Examples include the standard basis in R^n, where the basisindices are 1, 2, ..., n, or a Fourier basis for a function space, where indices correspond to frequency components. While not a distinct mathematical object, the notion of a basisindex is a practical labeling tool that supports notation, computation, and communication about basis-dependent representations.

See also: basis, coordinate vector, basis change, orthonormal basis.