3.14.4 Tensor Covector Basis Duality Preservation

Tensor Covector Basis Duality Preservation ensures consistent mapping between tensor and covector bases through dual space relationships.

Tensor Covector Basis Duality Preservation is the property that the defining pairing relation between a basis and its dual basis, namely that each dual basis functional returns 1 on its matching basis vector and 0 on all others, continues to hold after any change of basis, so that the new dual basis constructed from a new basis satisfies exactly the same Kronecker delta relation as the old one did. This preservation guarantees that duality is not a fragile relationship tied to one particular basis but a structural feature of the pairing between V and V* that survives every change of coordinates.

The Duality Relation Being Preserved

Statement for a Single Basis

For a basis e_1, ..., e_n of V, the dual basis e^1, ..., e^n of V* is defined by requiring:

e^{i} (e_{j}) = δ_{j}^{i}

where δ^i_j is 1 when i = j and 0 otherwise. This relation is what makes e^1, ..., e^n deserve the name dual basis, rather than simply being an arbitrary basis of V*.

What Preservation Asserts

Basis duality preservation asserts that if f_1, ..., f_n is any other basis of V, obtained from e_1, ..., e_n by a change-of-basis matrix A, and f^1, ..., f^n is constructed as the dual basis of f_1, ..., f_n, then the same relation holds:

f^{i} (f_{j}) = δ_{j}^{i}

The relation is not merely true again by a separate argument; it is preserved in the precise sense that the transformation rules for f_i and f^i are designed to guarantee this identity automatically.

Verifying Preservation Under the Transformation Rules

Recalling the Two Transformation Rules

The new basis vectors are given in terms of the old by f_i = A^j_i e_j, and the corresponding new dual basis functionals are given by f^i = (A^{-1})^i_k e^k, using the inverse matrix. These two rules were derived precisely so that the dual pairing relation would continue to hold.

Carrying Out the Verification

Substituting both expressions into f^i(f_j):

f^{i} (f_{j}) = \sum_{k = 1}^{n} \sum_{l = 1}^{n} {(A^{- 1})}_{k}^{i} e^{k} (A_{j}^{l}, e_{l})

Using e^k(e_l) = δ^k_l to collapse the double sum leaves a single sum over the shared index, which is exactly the matrix product of A^{-1} and A:

\sum_{k = 1}^{n} {(A^{- 1})}_{k}^{i} A_{j}^{k} = δ_{j}^{i}

since A^{-1}A is the identity matrix. The duality relation is recovered exactly, confirming preservation.

The Role of Using the Inverse for the Dual Basis

This computation shows precisely why the dual basis must transform with A^{-1} while the basis vectors transform with A: any other pairing of transformation matrices would fail to produce the identity matrix in the final step, and the Kronecker delta relation would not survive the change of basis. The inverse relationship between the two transformation rules is therefore not a coincidence but a requirement forced by duality preservation.

Consequence for Covector Components

Preservation Justifies the Component Transformation Law

Because the dual basis pairing relation is preserved across every change of basis, expressing a fixed covector φ in terms of any dual basis and using that dual basis's own pairing relation always yields a consistent set of components. This is what allows the earlier component transformation formula, φ'_i = A^j_i φ_j, to be trusted as producing components that behave correctly with respect to the new dual basis, since that dual basis satisfies the same defining relation as the old one.

Uniqueness of Components Guaranteed by Preservation

Given a covector φ and a basis, the duality relation e^i(e_j) = δ^i_j is exactly what makes the coefficients φ_i in the expansion φ = φ_i e^i uniquely determined, since pairing both sides with e_j isolates φ_j. Because the same relation is preserved for f_1, ..., f_n and f^1, ..., f^n, the new components φ'_i are equally well defined and equally unique, with no ambiguity introduced by the change of basis.

Preservation as a General Feature of Duality

Independence from the Particular Change of Basis Chosen

The verification above did not depend on any special property of the matrix A; it holds for any invertible matrix relating two bases of V. Duality preservation is therefore a universal feature of the dual basis construction, holding across every possible change of basis rather than for some restricted class of transformations.

Relation to the Evaluation Pairing's Invariance

Duality preservation at the level of basis vectors is the discrete, index-level counterpart of the more general fact that the evaluation pairing φ(v) is invariant under change of basis. The Kronecker delta relation is simply the evaluation pairing computed on basis vectors themselves, so its preservation is a special case, restricted to basis elements, of the broader invariance of evaluation described elsewhere in the theory of covectors.

Diagrammatic Summary

The diagram places the old and new duality pairings side by side, both equal to the same Kronecker delta, to emphasize that the change of basis in the middle carries the pairing relation across intact rather than disturbing it.