9.4.5 Tensor Product Basis Coordinate Expansion Role

Tensor Product Basis Coordinate Expansion Role is the part played by a tensor product basis in expressing any tensor of the corresponding type as a finite sum of basis elements weighted by scalar coefficients, so that the tensor is recovered exactly by adding together each basis element multiplied by its matching component; it is this expansion that gives concrete, computable meaning to the abstract tensor product basis, turning a set of basis elements into the mechanism by which any tensor can be reconstructed from a list of numbers.

The Expansion Itself

Every Tensor as a Weighted Sum

For a tensor product basis {e_i ⊗ f_j} built from factor bases {e_i} and {f_j}, the expansion role guarantees that any tensor T of matching type can be written as a sum running over every combination of basis indices, each term consisting of a coefficient T^{ij} multiplying the corresponding basis element.

No Term Is Optional

The expansion role requires that every basis element with a nonzero coefficient contribute its term to the sum, and that every basis element with a zero coefficient be safely omitted without altering the tensor recovered. No basis element may be left out of consideration when the expansion is being constructed, since omitting a term with a nonzero coefficient would produce a different tensor than the one being expanded.

Why the Expansion Role Is Necessary

Bridging Abstract Tensors and Concrete Numbers

A tensor, considered abstractly, is a multilinear map, not a list of numbers. The expansion role is what bridges this gap: it is the specific procedure that converts the abstract tensor into a finite, explicit combination of tensor product basis elements, and it is only through this procedure that a tensor can be manipulated using ordinary arithmetic on its coefficients.

Uniqueness of the Expansion

Given a fixed tensor product basis, the expansion role produces exactly one set of coefficients for any given tensor. No two distinct coefficient arrays expand to the same tensor relative to the same basis, and no tensor fails to have some coefficient array expanding to it, so the expansion role establishes a faithful, reversible correspondence between tensors and their coordinate arrays.

How the Expansion Role Interacts With Tensor Operations

Addition Adds Coefficients Termwise

When two tensors are added, the expansion role shows that the sum is obtained by adding, basis element by basis element, the coefficients of each tensor separately, since the tensor product basis elements themselves are fixed and only the weighting coefficients change under addition.

Scalar Multiplication Scales Every Coefficient

Multiplying a tensor by a scalar, viewed through the expansion role, scales every coefficient in its expansion by that same scalar while leaving the tensor product basis elements themselves untouched, since scalar multiplication distributes across the finite sum defining the expansion.

Diagram of the Expansion Role

Consequences of the Expansion Role

It Enables Termwise Verification of Tensor Identities

Because any tensor equation can be reduced, via the expansion role, to an equation between coefficients attached to the same basis elements on each side, identities between tensors can be verified or disproved by comparing coefficients term by term rather than by reasoning about the tensors abstractly.

It Is the Foundation for Component-Based Tensor Calculus

All subsequent component-based manipulation of tensors — contraction, raising and lowering of indices, and expression of tensor equations in coordinates — depends on the expansion role having already been carried out, since these manipulations operate directly on the coefficients that the expansion produces and have no meaning until that expansion is fixed.