5.11.3 Tensor Product General Factor Linearity

The tensor product's general factor linearity extends linear maps through multilinearity, linking algebraic structures and functional relationships.

Tensor Product General Factor Linearity is the unified statement, covering every slot at once rather than singling out the first or second factor, that for any index i between 1 and n, holding every factor other than the i-th fixed, the map sending a vector in Vi to the corresponding tensor is linear — the complete, symmetric formulation of which first factor linearity and second factor linearity are merely the two lowest special cases.

Formal Statement for an Arbitrary Slot

Fix vectors v1, …, vi-1, vi+1, …, vn in their respective spaces, one for every factor except the i-th. General factor linearity asserts that the map

L_{i} : V_{i} \to V_{1} \otimes \dots \otimes V_{n}, L_{i} (x) = v_{1} \otimes \dots \otimes v_{i - 1} \otimes x \otimes v_{i + 1} \otimes \dots \otimes v_{n}

is linear, for every choice of index i from 1 to n and every choice of the fixed vectors in the remaining n − 1 slots. This is the precise, complete content of the tensor product's multilinearity, stated once for all slots rather than as n separate special cases.

Relationship to the Special Cases

First and second factor linearity are the instances of this general statement obtained by setting i = 1 and i = 2 respectively; general factor linearity does not add mathematical content beyond what those and their analogues for i = 3, …, n already say collectively, but it packages them into a single quantified statement that avoids repeating the same argument n separate times.

Why a Single Unified Statement Matters

Stating the property once, quantified over i, rather than as n separate propositions, matters because most proofs and constructions in tensor algebra treat the slots interchangeably: an argument that reduces a general tensor's behavior to its behavior on basis vectors typically applies the same linearity argument to every slot in turn, and general factor linearity is the single lemma that licenses doing so uniformly rather than requiring n bespoke justifications.

Role in the Universal Property

General factor linearity is precisely the multilinearity condition appearing in the statement of the tensor product's universal property, and its full n-slot generality is essential to that statement's scope.

Multilinear Maps of Arbitrary Arity

The universal property asserts a bijection between linear maps out of V1 ⊗ ⋯ ⊗ Vn and multilinear maps out of V1 × ⋯ × Vn, where "multilinear" means precisely linear in every one of the n slots separately — the exact content of general factor linearity. Restricting attention to only the first or second slot, as in the special cases, would only characterize a weaker notion (partial linearity in one variable), insufficient to capture the full multilinear maps the universal property is built to classify.

Consistency Requirement Across All Slots

Because a multilinear map must satisfy the analogue of general factor linearity in every slot simultaneously, verifying that a candidate map qualifies as multilinear (and hence corresponds to a linear map on the tensor product) requires checking linearity in each of the n slots individually; missing even one slot's linearity disqualifies the map from the correspondence guaranteed by the universal property.

Inductive Verification Across Slots

General factor linearity for n factors can be established by induction on n, building on the two-factor case established directly from the construction of the tensor product.

Base Case and Inductive Step

The base case, n = 2, is verified directly from the bilinearity built into the construction of V1 ⊗ V2. For the inductive step, viewing V1 ⊗ ⋯ ⊗ Vn+1 as (V1 ⊗ ⋯ ⊗ Vn) ⊗ Vn+1 via the associativity isomorphism, linearity in any of the first n slots follows from the inductive hypothesis applied to the grouped factor V1 ⊗ ⋯ ⊗ Vn, while linearity in the final slot follows from the base two-factor bilinearity applied to the pair (V1 ⊗ ⋯ ⊗ Vn, Vn+1).

Slot Independence Preserved Under Associativity

Because the associativity isomorphism used at each inductive step is itself canonical and does not favor any particular grouping of factors, the resulting general factor linearity, once established inductively, holds symmetrically across all n slots and does not depend on the particular order in which the induction happened to group the factors.

Illustrative Diagram

The dashed box marks an arbitrary chosen slot among the n factors, illustrating that general factor linearity holds regardless of which particular slot is singled out to vary.