7.1.1 Tensor Component Representation Scope

Tensor Component Representation Scope is the delimitation of what counts as "the component representation" of a tensor — namely, the complete indexed array of all its components together with an explicit record of the basis used to produce them — as distinct from a single component in isolation on one side, and the abstract basis-free tensor on the other. Where component scope draws the line around a single scalar entry, representation scope draws a wider line around the entire array-plus-basis package, and clarifying this wider boundary is what prevents the common shorthand "the matrix of T" from being mistaken for a basis-free description of T itself.

What the Representation Includes

The Full Array, Not a Partial One

The component representation of a tensor T of type (p, q) includes every one of its N = d^{p+q} components, indexed completely; an array that has been partially filled in, or that reports only a subset of components (such as only the diagonal entries of a matrix), falls short of being the full representation and lies outside this scope until completed.

The Basis Data, Not Just the Numbers

A bare numerical array, presented without any statement of which bases were used to produce it, is not a complete component representation in the scope intended here — it is merely a table of numbers whose relationship to any specific tensor is undetermined until the missing basis information is supplied.

What Falls Outside the Representation's Scope

The Basis-Free Tensor Itself

The tensor T, as an element of the abstract space V* ⊗ W (or wherever it lives), is not itself part of any one component representation; the representation is a presentation of T relative to a choice, and T persists as the same object across every possible choice of representation, standing outside the scope of any single one of them.

Other Representations of the Same Tensor

A second component representation of the same T, computed in a different basis, is not "part of" the first representation, even though the two are related by the transformation law; each representation is a self-contained package (array plus its own specific basis data), and the scope of one representation does not automatically extend to include any other, related though they may be.

Diagram of Representation Scope

Why This Scope Distinction Is Useful

Preventing Silent Loss of Basis Information

Recognizing that a valid component representation must include its basis data prevents a common practical error: recording or transmitting a tensor's numerical array without also recording which basis produced it, after which the array becomes uninterpretable as a representation of any specific tensor, since the transformation law needed to relate it to any other basis can no longer be applied.

Clarifying What "Converting Between Representations" Means

Because the scope of a representation includes its basis data, "converting" a tensor from one representation to another is understood precisely as applying the transformation law to move from one full (array, basis) package to another, producing a new representation within its own separate scope, rather than modifying or extending the original representation in place.