9.3.2 Tensor Basis Ordering Rule

Tensor Basis Ordering Rule is the requirement that a basis used for tensor components be treated not merely as an unordered set of linearly independent spanning vectors but as an ordered list, since the specific order assigned to the basis vectors is what fixes which numerical index label attaches to which vector, and reordering the list — even though the underlying set of vectors and the space they span are unchanged — constitutes its own particular change of basis with its own Jacobian, namely a permutation matrix. It is the often-implicit convention, underlying every use of numerical indices, that turns an abstract spanning set into the specific ordered correspondence that index notation requires.

Why an Order Must Be Fixed

Indices Presuppose a Sequence, Not Just a Set

Writing e₁, e₂, ..., eₙ already assumes a definite first, second, and so on assignment among the basis vectors; the set {e₁, e₂, e₃} alone, without a specified order, does not determine which vector should be called e₁ rather than e₂, so index notation's entire labeling scheme depends on this ordering having been fixed in advance, separately from the more basic requirement that the vectors merely be linearly independent and spanning.

The Dual and Induced Bases Inherit the Same Order

Once an order is fixed for the primal basis, the dual basis and every induced tensor product basis inherit a matching order automatically, since the dual basis vector eⁱ is defined relative to the primal vector eᵢ occupying the i-th position; the basis ordering rule therefore governs the labeling of the entire basis system structure at once, not merely the primal basis in isolation.

Reordering as a Change of Basis by a Permutation Matrix

A Permutation Is a Special Case of a Change of Basis

Reassigning which vector is called e₁ versus e₂ (while keeping the same underlying vectors) is itself a change of basis, described by a permutation matrix — a matrix with exactly one entry equal to 1 in each row and column and all other entries zero — and tensor components transform under this reordering exactly as they would under any other change of basis, using this permutation matrix in place of a general Jacobian.

with P a permutation matrix, showing that reordering the basis is formally a basis change belonging to the global linear change of basis area, specialized to this particular kind of transformation matrix.

Numerical Components Change Even Though No Geometric Content Changes

Because reordering is a genuine change of basis, the numerical value attached to a specific index, such as v¹, generally changes when the basis order is altered, even though the underlying vector v and the underlying set of basis vectors are entirely unchanged; this is a reminder that individual components are fully basis-dependent in the sense described elsewhere, with ordering being one further respect in which that dependence manifests.

Consequences of Ordering for Orientation and Determinant Sign

Swapping Two Basis Vectors Flips the Sign of the Determinant

Because a permutation matrix that swaps two basis vectors has determinant −1, reordering a basis by such a swap flips the sign of the determinant of the transformation relating it to the original ordering; this is the direct source of the orientation-dependence encountered in objects such as the Levi-Civita symbol and in pseudotensor transformation behavior, both of which are sensitive specifically to whether a basis reordering is even or odd.

Orientation as an Equivalence Class of Orderings

Two orderings of the same underlying set of basis vectors are said to induce the same orientation if the permutation relating them has determinant +1, and the opposite orientation if the determinant is −1; the basis ordering rule is what makes this notion of orientation meaningful at all, since orientation is fundamentally a statement about equivalence classes of orderings rather than about the unordered spanning set itself.

Diagram of Reordering as a Permutation Change of Basis

Practical Implications of the Ordering Rule

Always Fixing an Order Before Assigning Indices

Because index notation is meaningless without a fixed order, any specification of a basis intended for use with tensor components must state, explicitly or by clear convention, which vector is meant by each index label; omitting this specification leaves the correspondence between indices and vectors genuinely ambiguous, independent of whether the underlying spanning set itself has been correctly identified.

Tracking Orientation Changes Across a Derivation

Whenever a derivation reorders basis vectors partway through — for instance, to align an index convention with a different but equally valid ordering used elsewhere — tracking whether the reordering is even or odd is necessary to correctly propagate the sign of any orientation-sensitive quantity, such as a volume form or the Levi-Civita symbol, through the remainder of the calculation.