9.3.5 Tensor Basis Coordinate Assignment Role

Tensor Basis Coordinate Assignment Role is the function that a chosen basis and its dual basis perform in converting an abstract tensor into a definite, ordered array of numbers — its coordinates — by pairing each basis or dual-basis element with the tensor through contraction, so that every slot of the tensor receives a numerical label tied to a specific basis vector or covector; this role is what makes coordinates possible at all, since a tensor by itself carries no numbers, only a basis assignment produces them, and changing the basis changes the numbers produced while leaving the tensor itself unchanged.

What the Assignment Role Actually Does

From Abstract Object to Labeled Array

A tensor is defined independently of any basis, as a multilinear object acting on vectors and covectors. The basis coordinate assignment role is the operation that takes this basis-free object and extracts from it a specific array of scalars by evaluating the tensor on the members of a chosen basis {eᵢ} and dual basis {eʲ}. Each evaluation fixes one entry of the component array, so the full array is produced only once every basis and dual-basis element has been paired against the tensor in turn.

Assignment Is Slot-by-Slot

The assignment role acts independently on every argument slot of a tensor. A slot that accepts vectors is assigned coordinates by pairing it with dual-basis covectors, while a slot that accepts covectors is assigned coordinates by pairing it with the primal basis vectors. This slot-by-slot pairing is what produces the distinction between upper and lower indices on the component array: each upper index marks a slot assigned through the dual basis, and each lower index marks a slot assigned through the primal basis.

Why the Role Is Basis-Dependent

No Coordinates Without a Choice

The assignment role cannot be performed without first fixing a basis. A tensor has exactly one identity as an abstract multilinear object, but it has as many coordinate arrays as there are bases available to assign them, because each basis pairs with the tensor differently. The role is therefore not a property of the tensor alone but a relation between the tensor and a chosen basis.

Consistency Under Change of Basis

When the basis is replaced by another, the assignment role must be re-performed with the new basis and its new dual, producing a new array of coordinates related to the old one by the standard transformation law. The role guarantees that this replacement is always well defined: given any basis, the pairing procedure always produces a unique coordinate for every slot, and given the transition data between two bases, the new coordinates are always recoverable from the old ones without any additional information.

The Two Halves of the Assignment Mechanism

The Primal Basis Assigns Lower Coordinates

The primal basis {eᵢ} is paired against covector slots of a tensor, and the resulting numbers are recorded as lower indices. This half of the assignment role captures how a tensor treats each individual basis vector, one direction at a time.

The Dual Basis Assigns Upper Coordinates

The dual basis {eʲ}, defined by the pairing eʲ(eᵢ) = δʲᵢ, is paired against vector slots of a tensor, and the resulting numbers are recorded as upper indices. This half of the assignment role captures how a tensor treats each individual covector direction, again one at a time. Together, the two halves account for every slot a tensor may have, regardless of tensor order or the mixture of vector and covector arguments.

Diagram of the Assignment Role

Consequences of This Role Within Tensor Basis Systems

It Grounds the Distinction Between Tensor and Component Array

Because the assignment role is the sole mechanism producing coordinates, it is what justifies treating a tensor and its component array as related but distinct objects: the tensor is invariant, while the component array produced by the assignment role is tied to whichever basis was used to perform the pairing. Any statement about "the coordinates of a tensor" implicitly presupposes that this assignment has already been carried out relative to some stated basis.

It Underwrites Every Later Operation on Coordinates

Operations such as raising and lowering indices, contracting tensors, or checking symmetry properties on a component array are only meaningful because the assignment role has already fixed, for that array, exactly which basis vector or covector each index refers to. Without a settled assignment, the numbers in an array carry no attachment to any geometric or algebraic direction, and none of these later operations could be interpreted correctly.