basisdepends

Basisdepends is a term used in mathematics and computer science to denote the study of how representations of entities depend on the chosen basis in a finite-dimensional vector space. It focuses on the relationships between different bases and the coordinate transformations that connect them.

Definition: Let V be an n-dimensional vector space over a field F. B = {b1, ..., bn} and C = {c1, ..., cn} are two bases. The coordinates of a vector v with respect to B and C are related by the change-of-basis matrix P from C to B, defined by [v]_B = P [v]_C for all v. P is invertible, and P^{-1} is the change from B to C. The study of basisdepends encompasses the properties of P, its conditioning, sparsity, and how basis choices affect computational tasks.

Rationale and applications: Many linear-algebra computations, such as solving systems, eigenvalue problems, or performing transforms, have different representations depending on the basis. Basisdepends provides a framework to compare representations, develop basis-aware algorithms, and understand how numerical error propagates through basis changes.

Examples: In R^2, with B = {e1, e2} and C = {f1 = e1+e2, f2 = e1−e2}, the change-of-basis matrix from C to B is P = [[1, 1], [1, −1]], yielding [v]_B = P [v]_C for any v.

Relation: Basisdepends relates to standard concepts of change of basis, coordinate systems, and basis independence; it emphasizes the dependence of coordinates on the chosen basis while invariants exist under basis changes.

See also: basis, change of basis, coordinates, linear transformation, vector space.