Basisindices

Basis indices refer to the labels attached to the vectors that form a basis of a vector space. In linear algebra, a basis B of a vector space V over a field F is a set of vectors that is linearly independent and spans V. If V has finite dimension n, any basis contains n vectors. When a basis is written as B = {b_i : i ∈ I}, where I is an index set (often I = {1,2,...,n}), the index i is called a basis index and serves to label the corresponding basis vector b_i. In some texts or software, the term basisindices (as a single word) is used to refer to the labels of the basis vectors.

Coordinates and representation: For any v ∈ V, there exist unique scalars v_i ∈ F such that v = ∑_{i∈I} v_i b_i. The vector of coefficients (v_i)_{i∈I} is the coordinate vector of v relative to B. If I = {1,...,n}, this corresponds to the usual n-tuple of coordinates. The map sending each basis vector b_i to the standard unit vector e_i establishes an isomorphism between V and F^n.

Applications and usage: In computational linear algebra and optimization, basis indices identify the columns or vectors currently forming a basis for a subspace. In the simplex method for linear programming, the current basis is a set of column indices B, and these B-indices are the basis indices. Pivot operations move indices between B and the non-basis set, updating representations accordingly.

Notes: The term basis indices is descriptive and emphasizes labeling of basis elements; basisindices is an alternate spelling found in some texts or software. The underlying idea—labeling basis vectors to index coordinates—appears across linear, tensor, and optimization contexts.