5.15.1 Tensor Product Additivity Relation

The Tensor Product Additivity Relation shows how tensor products distribute over direct sums, key in tensor algebra and mathematical structures.

Tensor Product Additivity Relation is the specific generator of the tensor product's defining equivalence relation that forces the canonical map to distribute over vector addition in each argument. It appears as two distinct identities, one for the left factor and one for the right factor, and together they form the additive half of the four relations, alongside scalar compatibility, that convert the free module on pairs into the tensor product proper.

The Two Additivity Identities

Left Additivity

For u, v ∈ V and w ∈ W, the left additivity relation states:

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

Right Additivity

For v ∈ V and w, x ∈ W, the right additivity relation states:

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

Realizing Additivity in the Quotient Construction

The Generating Elements of the Relation Submodule

Within the free module F(V × W), the additivity relation is enforced by including in the relation submodule R every element of the form:

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

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

Passing to Equivalence Classes

Setting these generators to zero in the quotient F(V × W)/R means precisely that the equivalence class of (u+v, w) coincides with the sum of the equivalence classes of (u, w) and (v, w), which is exactly the left additivity relation restated in terms of tensor notation.

Why Two Separate Identities Are Needed

Independence of the Two Arguments

Additivity must be imposed separately in each argument because the canonical map is bilinear, not linear, over the direct sum V ⊕ W. A single combined additivity condition covering both arguments simultaneously would be strictly stronger than bilinearity and would force V ⊗ W to collapse into something isomorphic to a much smaller space, destroying the multiplicative dimension count that the tensor product is meant to have.

Consequence If Only One Additivity Relation Held

If, hypothetically, only left additivity were imposed while right additivity were omitted, the resulting quotient would not be independent of decomposition in the second argument, and the induced structure would fail to be well-defined as a target for genuinely bilinear maps that are additive in both slots.

Extended Consequences of Additivity

Additivity Extends to Finite Sums

By induction, the additivity relations extend from two terms to any finite number of terms:

(\sum_{i = 1}^{n} v_{i}) \otimes w = \sum_{i = 1}^{n} v_{i} \otimes w

This extended additivity is the basis for expanding tensors of sums of basis vectors into sums of basis tensors, which underlies all coordinate computations with tensors.

Additivity and the Zero Element

Combined with the additive identity element, additivity implies 0 ⊗ w = 0 and v ⊗ 0 = 0, since 0 ⊗ w = (0 + 0) ⊗ w = 0 ⊗ w + 0 ⊗ w forces 0 ⊗ w = 0 after subtracting one copy from both sides.

Visual Illustration

Role Within the Broader Equivalence Structure

One Quarter of the Full Relation Set

The additivity relation constitutes two of the four generating families that define the relation submodule R, the remaining two being the scalar compatibility relations. Together, all four generators fully determine the equivalence relation whose quotient produces V ⊗ W.

Foundation for Multilinear Additivity

The same left-and-right additivity pattern generalizes to multilinear maps of n arguments, where additivity must be imposed independently in each of the n slots, producing n separate families of additivity generators in the corresponding relation submodule for the iterated tensor product.