3.9.4 Tensor Dual Coordinate Transformation Rule

The Tensor Dual Coordinate Transformation Rule describes how dual tensors transform under coordinate changes, maintaining their algebraic structure in multilinear spaces.

Tensor Dual Coordinate Transformation Rule is the precise matrix formula governing how the component list of a fixed covector changes when the basis of V is replaced by another, stated as a group action of the general linear group on coordinate tuples that is consistent under composition of successive changes of basis. Beyond simply stating that covector coordinates transform "with" the basis matrix, the transformation rule specifies the exact matrix involved, proves that composing two changes of basis behaves correctly, and explains why the dual basis itself transforms inversely to compensate.

Statement of the Transformation Rule

Change of Basis for V

Let e_1, ..., e_n and e'_1, ..., e'_n be two bases of V, related by an invertible matrix A in GL(n, F) via

e_{j'} = A_{j}^{i} e_{i}

The Induced Transformation on Covector Coordinates

For a fixed covector f with coordinates f_i = f(e_i) in the old basis and f̃_j = f(e'_j) in the new basis, the transformation rule states

{\tilde{f}}_{j} = A_{j}^{i} f_{i}

which, written as matrix multiplication of the row vector of coordinates by the transpose of A, reads f̃ = f A, treating f as a row vector multiplied on the right by the matrix A.

The Compensating Transformation of the Dual Basis

Why the Dual Basis Must Transform Inversely

Since the coordinate f̃_j is defined as f(e'_j), and the dual basis is characterized entirely by its pairing with the primal basis, consistency requires the new dual basis e'^1, ..., e'^n to satisfy e'^i(e'_j) = δ^i_j. Writing e'^i = (A^{-1})^i_k e^k, a direct check confirms

e^{i'} (e_{j'}) = {(A^{- 1})}_{k}^{i} e^{k} (A_{j}^{l} e_{l}) = {(A^{- 1})}_{k}^{i} A_{j}^{l} δ_{l}^{k} = δ_{j}^{i}

confirming the new dual basis is exactly the inverse-transpose transform of the old one, and this compensating relationship is what forces the covariant transformation law on coordinates to be consistent.

Consistency Under Composition

The Rule Defines a Genuine Group Action

If a second change of basis, given by matrix B, is applied after the first, moving from e'_1, ..., e'_n to e''_1, ..., e''_n, the transformation rule applied twice gives

{\tilde{\tilde{f}}}_{k} = B_{k}^{j} {\tilde{f}}_{j} = B_{k}^{j} A_{j}^{i} f_{i} = {(B, A)}_{k}^{i} f_{i}

matching exactly what would be obtained by applying the single combined change of basis with matrix BA directly. This composition property confirms the transformation rule defines a consistent right action of GL(n, F) on the space of coordinate lists, a necessary condition for the rule to be mathematically sound rather than an artifact valid only for a single change of basis.

Identity Transformation

Taking A to be the identity matrix leaves every coordinate unchanged, f̃_i = f_i, confirming the transformation rule correctly reduces to the trivial case when no actual change of basis occurs.

Relation to Vector Coordinate Transformation

The Paired, Opposite Rule

Vector coordinates transform according to the paired rule ṽ^i = (A^{-1})^i_j v^j, using the inverse of the same matrix A. The two rules are designed specifically so that the contracted product f̃_i ṽ^i equals f_i v^i for every choice of A, guaranteeing the evaluated scalar f(v) remains invariant regardless of which transformation rule is applied.

General Pattern for Higher-Rank Tensors

The same style of transformation rule extends to tensors of arbitrary type (p, q): each upper index transforms with a copy of A^{-1}, and each lower index transforms with a copy of A, and the full transformation law is simply the tensor product of these individual index transformations applied simultaneously to every index of the tensor's component array.

Diagrammatic Summary

The diagram illustrates that applying the transformation rule twice, once with A and once with B, matches applying it once with the combined matrix BA, confirming the rule's internal consistency.