basisdependent

Basisdependent is an adjective used to describe a quantity, representation, or property that relies on a particular choice of basis in a vector space or coordinate system. It is contrasted with basis-independent (or coordinate-free) notions, which remain the same under a change of basis.

In linear algebra, a finite-dimensional vector space V with a basis B = {b1, ..., bn} assigns to every vector x coordinates [x]_B that depend on B. A linear operator A on V is represented by a matrix [A]_B relative to B; when the basis changes to C, related by an invertible transition matrix P, the matrix transforms as [A]_C = P [A]_B P^{-1}, and coordinates transform as [x]_C = P [x]_B. Thus the numerical matrix and coordinate components are basisdependent, even though the underlying operator and its eigenvalues are generally basis-independent.

Some quantities are invariant under a basis change. These basis-independent properties include the spectrum of A, its trace, and its determinant. Conversely, many representations and objects are basisdependent: the Gram matrix G = [<b_i, b_j>] of a basis, the coordinate form of a metric, or the explicit matrix of a linear operator in a given basis.

Practical implications include the ability to simplify representations by choosing convenient bases (for example, an eigenbasis to diagonalize a matrix) while recognizing that such simplifications are basis-dependent. Awareness of basis dependence is important in numerical computation, where representation, conditioning, and sparsity can vary with the chosen basis.