3.9.2 Tensor Dual Coordinate Basis Dependence

Tensor Dual Coordinate Basis Dependence is the precise description of how the numerical coordinates assigned to a fixed covector change when the underlying basis of V is replaced by a different one, capturing both the exact transformation formula and the conceptual reason that coordinates, unlike the covector itself, are never basis-independent quantities. While the covector f remains one single, unchanging object, its coordinate tuple (f_1, ..., f_n) is tied entirely to whichever basis was used to construct the dual basis through which those coordinates were extracted.

The Source of Basis Dependence

Coordinates Are Defined Relative to a Basis

By construction, the coordinate f_i of a covector f is defined as f(e_i), the value of f on the i-th vector of a chosen basis. Since this definition explicitly references the basis vectors e_i, replacing the basis with a different one necessarily changes which numbers are produced, even though f itself has not changed as an abstract functional.

A Minimal Illustration

Consider V = R^2 and a fixed covector f with f(x, y) = x + y. Relative to the standard basis e_1 = (1, 0), e_2 = (0, 1), the coordinates are f_1 = f(e_1) = 1 and f_2 = f(e_2) = 1. Relative to a different basis e'_1 = (1, 1), e'_2 = (1, -1), the coordinates become f(e'_1) = 2 and f(e'_2) = 0, a completely different pair of numbers describing the exact same covector.

The Transformation Formula

Setting Up the Change of Basis

Let a new basis e'_1, ..., e'_n relate to the old basis by e'_j = A^i_j e_i for an invertible matrix A. Since f is linear,

so the new coordinates f̃_j are obtained from the old coordinates f_i by exactly the same matrix A used to express the new basis in terms of the old one. This is the covariant transformation law already used to explain the name of covariant components.

Reversibility of the Transformation

Since A is invertible, the transformation can always be undone: given the coordinates relative to the new basis, the coordinates relative to the old basis are recovered using A^{-1}, confirming that the basis dependence is a full, reversible change of description rather than a loss of information.

Contrast with Basis-Independent Quantities

Coordinates Versus the Covector Itself

The covector f as an abstract linear functional is a single, fixed mathematical object that does not change when a basis is chosen or altered. Only its coordinate description, the tuple of numbers used to compute with it, depends on the basis. This distinction, between an invariant object and its variant description, is one of the central organizing ideas of tensor algebra.

Coordinates Versus Evaluation Results

Although the individual coordinates f_i and v^i both depend on the basis, the scalar f_i v^i obtained by evaluating f on a vector v does not, since the transformation of f_i by A and the transformation of v^i by A^{-1} cancel exactly. Basis dependence in coordinate descriptions and basis independence in evaluated scalars coexist consistently within the same framework.

Practical Implications

Coordinates Must Always Be Stated Relative to a Basis

Any numerical coordinate description of a covector is meaningless without specifying which basis it is relative to; the same tuple of numbers could describe entirely different covectors depending on the basis assumed. Careful bookkeeping of which basis is in use is therefore essential whenever coordinate representations of covectors are compared or combined.

Detecting Errors Through Basis Dependence

A useful diagnostic when checking a proposed formula involving covector components is to verify that it transforms correctly under a change of basis using the covariant rule; a formula that fails to transform this way is likely combining components inconsistently, such as mistakenly summing two covariant indices without a compensating contravariant partner.

Diagrammatic Summary

The diagram shows one fixed covector producing two different coordinate tuples depending on which basis is used, illustrating the basis dependence of coordinate description.