9.8 Tensor Basis Tensor Role

Tensor Basis Tensor Role is the recognition that each element of a tensor basis is not merely a labeling device external to the tensors it helps describe, but is itself a genuine tensor, belonging to the very tensor space it is used to span, so that basis vectors, dual basis covectors, and their tensor products can themselves be added, scaled, contracted, and paired with other tensors exactly like any other tensor of matching type; it clarifies that a basis is drawn from within the space of tensors rather than supplied from some separate external source.

Basis Elements as Full Members of Their Tensor Space

A Basis Vector Is a Rank-One Tensor

Each primal basis vector e_i is itself an order-one tensor, satisfying every property required of a vector: it can be added to other vectors, scaled by a scalar, and paired with covectors to produce a number, exactly as any other vector in the space would be.

A Dual Basis Covector Is Equally a Genuine Tensor

The dual basis covectors e^i are, in exactly the same way, full members of the dual space, capable of being combined with other covectors and of accepting vectors as arguments to produce scalars, with no special status distinguishing them from any other covector in that dual space.

Tensor Product Basis Elements Are Higher-Order Tensors

Products of Basis Elements Belong to the Higher-Order Tensor Space

A tensor product such as e_i ⊗ e^j, formed from basis and dual-basis elements, is itself an order-two tensor, belonging fully to the space of mixed tensors it helps to span; it can be added to any other tensor of that type and can accept the appropriate arguments to produce a scalar, just as any other rank-two tensor could.

This Role Is What Makes Basis Expansion Consistent

Because tensor product basis elements are themselves tensors of the correct type, forming a weighted sum of them to expand an arbitrary tensor produces another object of the same type, rather than an object belonging to some auxiliary structure; this consistency is what allows the expansion role of a tensor basis to make sense as an equation between tensors on both sides.

Distinguishing the Tensor Role From the Assignment Role

The Assignment Role Uses Basis Elements as Tools

When a basis is used to assign coordinates, its elements are treated instrumentally, as the means by which a tensor's slots are evaluated to produce numbers; this use does not by itself require recognizing the basis elements as tensors, only as objects capable of being paired with tensor slots.

The Tensor Role Recognizes What the Basis Elements Are

The tensor basis tensor role goes further, asserting that the objects being used this way are not external tools at all but are themselves ordinary tensors, meaning any general fact proved about tensors of a given type applies automatically to the basis elements of that same type.

Diagram of the Tensor Role

Consequences of the Tensor Basis Tensor Role

General Tensor Theorems Apply Directly to Basis Elements

Because basis elements are themselves tensors, any theorem established for tensors in general — regarding linear combinations, contraction, or transformation under a change of basis — applies without modification to the basis elements as a special case, requiring no separate proof tailored specifically to them.

It Clarifies Why a New Basis Can Be Built From Combinations of an Old One

Recognizing basis elements as genuine tensors is what justifies constructing an entirely new basis by taking linear combinations of an existing basis's elements: since the combinations are themselves ordinary tensors of the correct type, they are eligible, subject to remaining linearly independent, to serve as a basis in their own right.