5.16.1 Tensor Product Formal Generator

Tensor Product Formal Generator is a single simple tensor v ⊗ w, viewed specifically in its role as a generating element of the vector space V ⊗ W, meaning one of the building blocks whose finite linear combinations produce every element of the tensor product. Calling v ⊗ w a "generator" emphasizes its structural function within the spanning set of V ⊗ W, distinguishing this role from its role as an output of the canonical map or as a rank-one tensor, even though all three descriptions refer to the same underlying object.

Generators as a Spanning Family

The Full Set of Generators

The set of all formal generators is:

and by definition of the tensor product, this set spans V ⊗ W as a vector space:

Generators Are Redundant, Not a Basis

Unlike a basis, the set G of formal generators is highly redundant: it is generally infinite even when V ⊗ W is finite-dimensional, and many distinct generators, such as v ⊗ w and (cv) ⊗ (c⁻¹w), coincide as elements of the tensor product due to homogeneity.

Reducing Generators to a Basis

Basis Generators from Basis Vectors

If {eᵢ} and {fⱼ} are bases for V and W, the subset of generators {eᵢ ⊗ fⱼ} forms an actual basis of V ⊗ W, not merely a spanning set. Every other generator v ⊗ w can be expanded, using additivity and homogeneity, as a linear combination of these basis generators:

where v = Σ aᵢeᵢ and w = Σ bⱼfⱼ are the coordinate expansions of v and w in their respective bases.

From Spanning Generators to a Minimal Basis

This shows that the full family of generic formal generators, though it spans V ⊗ W, contains within it a much smaller finite basis obtained by restricting to basis-vector pairs, and it is this minimal basis that gives the exact dimension count dim(V) · dim(W).

Verifying Constructions on Generators Alone

Defining Linear Maps by Their Effect on Generators

Because the formal generators span V ⊗ W, a linear map out of the tensor product is completely determined once its values on all formal generators v ⊗ w are specified, consistent with the bilinearity constraints. This is the practical technique used throughout tensor algebra: define a map "on simple tensors" and extend by linearity.

The Consistency Requirement

Because the generating set G is redundant, specifying values on generators requires care: the assignment v ⊗ w ↦ β(v, w) extends to a well-defined linear map on V ⊗ W if and only if β is bilinear, ensuring the assignment is consistent across all of the redundant relations among generators.

Diagram of Generators Spanning the Space

Generators in the Free Module Construction

Origin of the Term "Generator"

The terminology "generator" originates directly from the free module F(V × W), in which each pair (v, w) is literally a free generator of the module, with no relations imposed. After passing to the quotient F(V × W)/R, the images of these free generators become the formal generators v ⊗ w of V ⊗ W, retaining their generating role even though they are no longer free.

Loss of Freedom, Retention of Spanning

While the free generators of F(V × W) satisfy no relations and hence form a genuine basis of the free module, the formal generators of V ⊗ W satisfy the bilinear relations and no longer form a basis in general, illustrating precisely how the quotient construction trades linear independence for the additional bilinear structure.

Broader Significance

Template for Generator-Based Reasoning

Reasoning about V ⊗ W via its formal generators mirrors a broader technique in algebra where a structure defined by a quotient of a free object is analyzed by understanding how its generators behave under the imposed relations, a method used equally for group presentations, quotient rings, and other quotient constructions.

Extension to Iterated Tensor Products

The same generator concept extends to n-fold tensor products, where the formal generators are simple tensors v₁ ⊗ v₂ ⊗ ... ⊗ vₙ, and a basis is again obtained by restricting to all combinations of basis vectors from each of the n factor spaces.