5.12.5 Tensor Product Bilinear Compatibility

Tensor Product Bilinear Compatibility is the requirement, and the resulting guarantee, that constructions built on top of the tensor product's bilinear map — most importantly the induced maps obtained by tensoring together linear maps on the individual factors — are well defined precisely because they respect the bilinearity relations that the tensor product itself is built to satisfy.

The Compatibility Requirement

Given linear maps f : V → V′ and g : W → W′, one wants to define an induced map f ⊗ g : V ⊗ W → V′ ⊗ W′ by declaring

on decomposable tensors and extending by linearity. Bilinear compatibility is the fact that this rule is consistent with every relation the ⊗ symbol satisfies — additivity and scalar compatibility in each slot — so that the definition does not run into contradictions when the same tensor is written as a decomposable product in more than one way, or when a sum of decomposable tensors is involved.

Verifying Compatibility with the Bilinear Relations

Establishing that f ⊗ g is well defined amounts to checking that the assignment v ⊗ w ↦ f(v) ⊗ g(w) respects each of the identities that define bilinearity of the tensor product.

Compatibility with Additive Distribution

Applying the proposed rule to (v1 + v2) ⊗ w gives f(v1 + v2) ⊗ g(w), which by linearity of f equals (f(v1) + f(v2)) ⊗ g(w), which by bilinearity of the tensor product in the target space equals f(v1) ⊗ g(w) + f(v2) ⊗ g(w) — matching the value obtained by applying the rule separately to v1 ⊗ w and v2 ⊗ w and adding, confirming compatibility with additive distribution in the first slot; the analogous check in the second slot uses linearity of g.

Compatibility with Scalar Distribution

Applying the proposed rule to (λv) ⊗ w gives f(λv) ⊗ g(w) = λf(v) ⊗ g(w) = λ(f(v) ⊗ g(w)), using linearity of f followed by scalar distribution in the target tensor product, matching λ times the value obtained on v ⊗ w directly — confirming compatibility with scalar distribution.

Universal Property as the Formal Justification

Rather than checking these identities by hand in every instance, the standard justification invokes the universal property directly: the composite map (v, w) ↦ f(v) ⊗ g(w) is a bilinear map from V × W into V′ ⊗ W′ (being a composition of the linear maps f and g with the bilinear tensor product map on the target), and the universal property guarantees a unique linear map V ⊗ W → V′ ⊗ W′ agreeing with it on decomposable tensors — this unique linear map is f ⊗ g, and its existence is the compatibility guarantee itself, supplied automatically rather than verified case by case.

Functorial Consequences of Compatibility

Once bilinear compatibility guarantees that f ⊗ g is well defined for any pair of linear maps, several further identities follow automatically, making tensoring of linear maps behave functorially.

Compatibility with Composition

holds because both sides agree on decomposable tensors by direct computation, and bilinear compatibility guarantees that agreement on decomposable tensors extends to agreement everywhere.

Compatibility with Identity Maps

The identity map on V ⊗ W is exactly idV ⊗ idW, again because both act identically on every decomposable tensor and bilinear compatibility extends this to the whole space; together with compatibility with composition, this identity is what makes "tensoring together" a functor from pairs of vector spaces and linear maps to vector spaces and linear maps.

Compatibility with Additional Structure

Bilinear compatibility extends beyond plain linear maps to structure-preserving maps of other kinds, whenever that structure interacts appropriately with the tensor product's bilinearity.

Compatibility with Inner Products

If V and W carry inner products, the tensor product ⟨·,·⟩V ⊗ ⟨·,·⟩W defined by ⟨v ⊗ w, v′ ⊗ w′⟩ = ⟨v, v′⟩⟨w, w′⟩ is well defined on decomposable tensors and extends bilinearly (in fact, sesquilinearly over the complex numbers) to the whole tensor product by the same compatibility argument used for linear maps, since the defining formula respects additive and scalar distribution in each factor.

Compatibility with Algebra Structures

When V and W are themselves algebras (vector spaces with a compatible multiplication), the tensor product V ⊗ W acquires an algebra structure with multiplication (v1 ⊗ w1)(v2 ⊗ w2) = (v1 v2) ⊗ (w1 w2), and bilinear compatibility is again what guarantees this multiplication is well defined on all of V ⊗ W, not merely on decomposable tensors, once the same additive and scalar consistency checks are carried out.

Illustrative Diagram

The induced map f ⊗ g is guaranteed to be a legitimate, well-defined linear map precisely because it respects every additive and scalar relation the tensor product's bilinear structure imposes.