3.4.5 Tensor Dual Basis Basis Dependence

Tensor Dual Basis Basis Dependence is the recognition that the dual basis {e^i} is not an intrinsic structure attached to V* alone, but a construction that exists only relative to a specific, prior choice of basis {e_i} for V, together with the consequences of this dependence: that different bases of V produce genuinely different dual bases, that no dual basis is privileged as canonical, and that any claim phrased in terms of "the" dual basis is implicitly relative to whichever basis of V was chosen at the outset.

No Canonical Dual Basis

Different Bases Yield Different Dual Bases

Given two distinct bases {e_i} and {e'_i} of V, related by a nontrivial transition matrix A ≠ I, the resulting dual bases {e^i} and {e'^i} are, in general, different sets of covectors; only in the trivial case A = I, meaning the two bases coincide, do the dual bases coincide as well. This is a direct manifestation of the absence of a natural isomorphism V ≅ V*, discussed in dual spaces and covectors: if a canonical dual basis existed independent of the choice of basis on V, it would supply exactly the missing natural identification.

Basis Dependence Is Not a Defect

This dependence is not a shortcoming of the dual basis construction but an accurate reflection of the underlying mathematics: V* genuinely has no distinguished basis unless V is given one first, in exactly the same way that V itself has no distinguished basis; the dual basis inherits whatever basis structure V is equipped with, and nothing more.

Illustrative Comparison of Dual Bases

A Concrete Pair of Bases

Take V = F^2 with the standard basis {e_1, e_2} = {(1,0), (0,1)}, whose dual basis is the standard dual basis {e^1, e^2} reading off the two coordinates directly, e^1(x, y) = x and e^2(x, y) = y. For the alternative basis {e'_1, e'_2} = {(2,1), (1,1)} used in tensor dual basis construction, the dual basis was computed there as e'^1(x, y) = x - y and e'^2(x, y) = -x + 2y, visibly different functionals from e^1 and e^2.

Neither Choice Is More Correct

Both {e^i} and {e'^i} are entirely valid dual bases, each correctly satisfying the defining relation relative to its own originating basis; there is no sense in which the standard dual basis is more fundamental than the alternative one, only that each is tied to the particular basis of V that produced it.

Consequences for Coordinate-Dependent Quantities

Covector Components Depend on the Chosen Dual Basis

Because a covector's coordinates ω_i = ω(e_i) are defined relative to a specific basis, and the dual basis itself changes when the underlying basis of V changes, the numerical components of a fixed covector ω differ across different choices of basis, following the covariant transformation rule already established; this is the coordinate-level manifestation of dual basis dependence, and it is why covector components always carry an implicit reference to whichever basis produced them.

Basis-Independent Quantities Remain Unaffected

Quantities built from full contractions between vectors and covectors, such as ω(v), are unaffected by the choice of underlying basis, since the basis-dependence of the dual basis and the basis-dependence of vector coordinates cancel exactly, as shown by the coordinate transfer cancellation Σ_k b^i_k a^k_j = δ^i_j; dual basis dependence at the level of individual coordinates is fully compatible with, and indeed is what guarantees, basis independence at the level of fully contracted scalar results.

Distinguishing Dual Basis Dependence From Domain Dependence

Two Separate Kinds of Dependence

Dual basis dependence, concerning the choice of basis on V, is distinct from the domain dependence discussed in tensor covector domain relation, concerning the choice of the vector space V itself: fixing V once and for all still leaves open which basis of V to use, and it is this residual freedom, after the domain has already been fixed, that dual basis dependence specifically addresses. A covector is tied to one domain V, but within that fixed domain, its dual basis expansion still depends on which particular basis of V was selected.

Practical Implication

When comparing dual basis expressions from different sources, or verifying a claimed tensor identity across different presentations, it is necessary to confirm that a common, explicitly stated basis of V underlies every dual basis expansion being compared; dual basis quantities computed relative to silently different bases of the same V are not directly comparable without first converting them, via the transition matrix, to a shared basis.