9.7.1 Tensor Standard Basis Element Ordering

Tensor Standard Basis Element Ordering is the convention that fixes a definite sequence for the vectors of the standard basis, listing e_1 before e_2, e_2 before e_3, and so on, matching the natural order of their indices, so that any array of components written down in the standard basis has an unambiguous correspondence between array position and basis vector; it is the specific, near-universal choice of ordering adopted for the standard basis, standing as a concrete instance of the more general ordered tuple role required of any tensor product basis.

What the Ordering Fixes

A Sequence Matching the Natural Number Order

The standard basis element ordering lists e_1, e_2, …, e_n in the same order as the natural numbers indexing them, so that no separate choice needs to be made about how to sequence the basis: the ordering is inherited directly and automatically from the ordering already present among the index values themselves.

Position in the Array Matches Position in the Sequence

Because of this ordering, the i-th entry of a vector's component array is understood, without further comment, to be the coefficient multiplying the i-th basis vector in this fixed sequence, so that array position and basis vector identity are kept in lockstep by the ordering alone.

Consistency Across Tensor Products of the Standard Basis

Multi-Index Ordering Inherits the Same Convention

When tensor products of the standard basis are formed to produce bases for higher-order tensors, the ordering of each factor in a multi index address follows the same natural-number sequence used for the single-factor standard basis, so that a basis element such as e_i ⊗ e_j is listed according to i and then j, each independently following the standard ordering.

Matrix and Array Layouts Depend on This Ordering

The familiar layout of a matrix, with rows and columns numbered in increasing order, is a direct consequence of applying the standard basis element ordering to both index positions of a rank-two tensor; changing the ordering convention would rearrange which entry of the matrix corresponds to which pair of basis vectors, without changing the tensor itself.

What Would Change Under a Different Ordering

The Same Tensor, a Different-Looking Array

If the standard basis vectors were listed in a different sequence — for instance, from e_n down to e_1 — the same tensor would be represented by an array with its entries permuted accordingly; the entries themselves, and the tensor they represent, would remain unchanged, only their arrangement in the array would differ.

Why the Natural Ordering Is Nearly Universal

The natural-number ordering is adopted almost universally for the standard basis because it requires no additional specification beyond the indices already used to label the basis vectors, and because it keeps array layouts consistent with the most familiar conventions for writing sequences of numbers.

Diagram of Standard Basis Element Ordering

Consequences of the Ordering Convention

It Allows Component Arrays to Be Compared Without Restating the Basis

Because the standard basis element ordering is fixed by convention, two component arrays produced in the standard basis can be compared entry by entry directly, without needing to restate which basis vector each entry corresponds to, since that correspondence is already fixed by position alone.

Any Deviation From the Convention Must Be Stated Explicitly

Should a nonstandard ordering of the standard basis vectors ever be used, that deviation must be stated explicitly alongside any component array produced under it, since readers otherwise default to assuming the natural-number ordering whenever the standard basis is named.