5.11 Tensor Product Linearity Property

Tensor Product Linearity Property is the defining requirement that the tensor product operation be linear in each of its slots separately, holding all other factors fixed, and the full set of algebraic identities — additivity and scalar compatibility in every argument — that this multilinearity (more precisely, in the two-factor case, bilinearity) entails throughout tensor algebra.

Statement of the Property

For vector spaces V1, V2, …, Vn over a field F, the tensor product map

is linear in each argument separately: fixing every input except the i-th, the resulting map from Vi into the tensor product is linear. Concretely, for the two-factor case this unpacks into two families of identities, additivity and scalar compatibility, holding independently in each of the two slots.

The Bilinear Identities in the Two-Factor Case

The two-factor tensor product satisfies four elementary identities that jointly constitute its bilinearity, each following directly from linearity in one slot at a time.

Additivity in the First Slot

Additivity in the Second Slot

Scalar Compatibility

These four identities together are precisely what is meant by describing the tensor product as bilinear, and the corresponding n-fold generalization, requiring exactly the analogous identity in each of the n slots independently, is what is meant by multilinearity.

Origin of the Property

Linearity in each slot is not an incidental convenience but is built directly into the construction of the tensor product, so that any construction failing to exhibit it would not qualify as a tensor product at all.

Built Into the Universal Property

The universal property defining the tensor product explicitly requires τ to be multilinear, and requires that every multilinear map out of V1 × ⋯ × Vn factor uniquely through τ; multilinearity of τ is therefore part of the very definition, not a derived consequence proved afterward.

Enforced by the Quotient Construction

In the explicit free-vector-space construction, the tensor product is obtained by quotienting by exactly the relations needed to force these additivity and scalar-compatibility identities to hold; the quotient is defined so that, for instance, (v1 + v2, w) and (v1, w) + (v2, w) become identified, guaranteeing additivity in the first slot by direct construction rather than by subsequent proof.

Consequences of the Linearity Property

The multilinearity of the tensor product operation is the single property from which the majority of standard tensor algebra identities are derived.

Distribution Over Sums

Because of additivity in each slot, expanding a tensor product of sums distributes term by term across all factors simultaneously, producing a sum with one term for every combination of summands chosen from each factor — the mechanism by which a general tensor's coordinate expansion is derived from the coordinate expansions of decomposable factors.

Non-Injectivity of the Map τ

Multilinearity, combined with scalar compatibility, is also the source of the tensor product's failure to be injective as a map from tuples to tensor product elements: because (λv) ⊗ w = v ⊗ (λw) for every scalar λ, distinct tuples can map to the same tensor, which is exactly the scalar-redistribution ambiguity affecting decomposable product expressions.

Foundation for the Universal Property's Bijection

The correspondence between multilinear maps on V1 × ⋯ × Vn and linear maps on the tensor product relies entirely on the linearity property: a linear map on the tensor product, precomposed with τ, is automatically multilinear because τ itself is multilinear, and conversely, any multilinear map can be shown to factor uniquely through τ precisely because τ's multilinearity captures every constraint such a factoring must satisfy.

Illustrative Diagram

The equality illustrates additivity in the first slot, one of the four elementary identities that together define the linearity property of the tensor product.