5.5.5 Tensor Elementary Expansion Role

The Tensor Elementary Expansion Role breaks down complex tensors into fundamental components, enabling clearer analysis and manipulation within algebraic structures.

Tensor Elementary Expansion Role is the function elementary tensors serve as the building blocks from which every element of V ⊗ W is assembled by finite summation, and the proof technique this expansion enables: establishing a fact for elementary tensors alone and then extending it to all of V ⊗ W by linearity.

Expansion as Spanning

Every Element Is a Finite Sum of Elementary Tensors

By construction, every element of V ⊗ W arises as the image of some element of the free vector space F(V × W) under the quotient map, and since F(V × W) is spanned by the formal symbols (v, w), every element of V ⊗ W is a finite sum ∑_{i=1}^{r} v_i ⊗ w_i of elementary tensors. This is the expansion referred to by the name of this topic: the writing of a general element as an explicit sum of elementary pieces.

Expansion Is Not Unique

The same element admits many different expansions — different values of r, different choices of v_i and w_i — related to one another by the bilinearity identities already fixed during construction; the expansion role of elementary tensors concerns the existence of at least one such expansion for every element, not the selection of a canonical or minimal one, which is instead the concern of tensor rank.

The Linearity-Extension Proof Technique

Verify on Elementary Tensors, Extend by Additivity

A recurring technique throughout tensor algebra is to define or verify a linear map, or an identity involving one, by checking it only on elementary tensors v ⊗ w and then invoking linearity to extend the result to arbitrary sums ∑ v_i ⊗ w_i. This works because a linear map is completely determined by its values on any spanning set, and the elementary tensors are exactly such a spanning set for V ⊗ W.

Why This Technique Is Sound

If two linear maps T_1, T_2: V ⊗ W → U agree on every elementary tensor, then for any general element t = ∑ v_i ⊗ w_i,

T_{1} (t) = T_{1} (\sum_{i} v_{i} \otimes w_{i}) = \sum_{i} T_{1} (v_{i} \otimes w_{i}) = \sum_{i} T_{2} (v_{i} \otimes w_{i}) = T_{2} (t)

by the linearity of T_1 and T_2 applied to the finite sum defining t; agreement on elementary tensors forces agreement everywhere.

Where This Technique Is Already Used

The uniqueness clause of the universal property is exactly an instance of this technique: two linear maps B̃_1, B̃_2 agreeing on every decomposable element v ⊗ w are shown equal by the same linearity-extension argument; the well-definedness of the induced map f ⊗ g and the coordinate expansion relative to a chosen basis both rely on the same reduction from an arbitrary element to elementary pieces.

Relation to Pure Tensor Structure

Expansion Presupposes Elementary Tensors, Not Their Minimality

The expansion role concerns only the existence of a sum of elementary tensors representing a given element, drawing on the same objects examined as pure tensors elsewhere; it does not address how few elementary tensors are needed, which is the separate question of tensor rank addressed under pure tensor structure and factorization.