5.12.3 Tensor Product Bilinear Relation Set

The tensor product bilinear relation set defines how tensors combine linearly, forming a foundational structure in multilinear algebra.

Tensor Product Bilinear Relation Set is the collection of algebraic identities that formally define how the tensor product operation ⊗ interacts with vector addition and scalar multiplication in the underlying vector spaces or modules. This set of relations is what makes the tensor product a genuinely bilinear construction rather than an arbitrary pairing of elements, and it serves as the defining property from which the universal mapping property of the tensor product is derived.

Definition of the Relation Set

The Core Identities

For vector spaces (or modules) V and W over a field (or ring) F, with u, v ∈ V, w, x ∈ W, and c ∈ F, the Tensor Product Bilinear Relation Set consists of the following four families of relations imposed on the free construction of V ⊗ W:

(u + v) \otimes w = u \otimes w + v \otimes w

u \otimes (w + x) = u \otimes w + u \otimes x

(c u) \otimes w = c (u \otimes w)

u \otimes (c w) = c (u \otimes w)

Distributivity in the Left Argument

The first identity states that the tensor product distributes over addition when the sum occurs in the left-hand factor. This mirrors ordinary distributivity of multiplication over addition, but here the "multiplication" is the abstract bilinear operation ⊗ rather than a numeric product.

Distributivity in the Right Argument

The second identity is the mirror image of the first: addition in the right-hand factor also distributes across the tensor product. Together, these two relations guarantee that ⊗ is additive in each slot independently.

Scalar Compatibility

The third and fourth identities establish that scalars can be freely moved across the tensor symbol, from either the left or right factor, without changing the resulting element. This scalar compatibility relation is essential for V ⊗ W to inherit a well-defined vector space (or module) structure over F.

Construction via Quotient of the Free Module

The Free Module on Pairs

The formal construction of V ⊗ W begins with the free module F(V × W) generated by all ordered pairs (v, w). This free module treats every pair as an independent basis element, with no relations imposed among them.

The Subspace of Relations

The Tensor Product Bilinear Relation Set is realized concretely as the submodule R generated by all elements of the form:

(u + v, w) - (u, w) - (v, w)

(u, w + x) - (u, w) - (u, x)

(c u, w) - c (u, w)

(u, c w) - c (u, w)

The Resulting Quotient

The tensor product space is then defined as the quotient:

V \otimes W = F (V \times W) / R

Every element of the free module that lies in R becomes zero after passing to the quotient, which is precisely what forces the bilinear relations to hold among the equivalence classes u ⊗ w, the image of (u, w) under the quotient map.

Role in the Universal Property

Bilinear Maps Factor Through the Tensor Product

The Tensor Product Bilinear Relation Set is exactly what is needed to guarantee the universal property of the tensor product: any bilinear map β: V × W → Z factors uniquely through a linear map V ⊗ W → Z. This works because β already satisfies the same relations that were quotiented out, so it descends unambiguously to the quotient space.

Uniqueness up to Isomorphism

Because the relation set is defined purely in terms of the bilinear identities, any two constructions of a tensor product satisfying this relation set are canonically isomorphic. The relation set therefore characterizes the tensor product independently of the specific model used to build it, whether via the free-module quotient, a basis-indexed construction, or a categorical colimit.

Consequences for Computation

Simple Tensors Are Not Closed Under Addition

A direct consequence of the relation set is that a general element of V ⊗ W is a finite sum of simple tensors u ⊗ w, since the relations only tell us how to manipulate sums and scalar multiples but do not collapse every sum of simple tensors into a single simple tensor.

Bilinear Relations Extend to Multilinear Settings

The same relation set generalizes directly to multiple factors, producing the defining relations for the multilinear tensor product V₁ ⊗ V₂ ⊗ ... ⊗ Vₙ, where distributivity and scalar compatibility must hold independently in each of the n argument slots.