5.14.3 Tensor Canonical Map Bilinearity

The tensor canonical map leverages bilinearity to construct tensor products, bridging linear maps and multilinear structures in algebra.

Tensor Canonical Map Bilinearity is the specific property possessed by the canonical map ⊗: V × W → V ⊗ W, guaranteeing that it is additive and scalar-compatible in each of its two arguments separately, without requiring the combined linearity that would be needed if V × W were treated as a single direct sum vector space. This bilinearity is not an incidental feature of the canonical map; it is the exact property that the tensor product's quotient construction is designed to enforce, and it is the reason simple tensors satisfy the algebraic manipulation rules used throughout multilinear algebra.

The Bilinearity Conditions

Additivity in the First Argument

For all u, v ∈ V and w ∈ W, the canonical map satisfies:

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

Additivity in the Second Argument

For all v ∈ V and w, x ∈ W:

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

Scalar Compatibility in Both Arguments

For every scalar c ∈ F:

(c v) \otimes w = c (v \otimes w) = v \otimes (c w)

Why Bilinearity Holds by Construction

The Relations Were Built to Force This

In the quotient construction of V ⊗ W, the submodule R of the free module F(V × W) is defined generated by exactly the elements that express the failure of additivity and scalar compatibility, such as (u+v, w) − (u,w) − (v,w). Quotienting by R sets each such difference to zero, which is precisely what makes the equivalence classes v ⊗ w obey the bilinearity conditions.

Bilinearity as a Design Requirement, Not a Theorem About an Arbitrary Map

Unlike properties that must be independently verified for a given map, the bilinearity of the canonical map is guaranteed automatically once V ⊗ W is defined as the quotient by the relation submodule. In this sense, bilinearity is closer to a specification the construction satisfies by design than a separate result requiring proof from unrelated axioms.

Consequences of Bilinearity

Zero Tensors from Zero Vectors

Bilinearity immediately implies:

0 \otimes w = v \otimes 0 = 0

since 0 ⊗ w = (0·v) ⊗ w = 0·(v ⊗ w) = 0 for any choice of v, and similarly for the second argument.

Sign and Scalar Migration

Bilinearity also permits moving scalars, including negative signs, freely between the two factors:

(- v) \otimes w = - (v \otimes w) = v \otimes (- w)

which justifies the standard rule that a scalar factor can be relocated across the tensor symbol without changing the resulting element.

Distinguishing Bilinear from Linear Behavior

Bilinearity does not imply that ⊗ is additive across the combined pair, meaning:

(u + v) \otimes (w + x) \neq u \otimes w + v \otimes x

in general. Instead, expanding this expression by applying the two bilinearity rules in sequence produces four separate terms, not two.

Full Bilinear Expansion Example

Expanding a Sum of Sums

Applying additivity in both arguments to (u+v) ⊗ (w+x) gives:

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

Illustration of the Expansion

Broader Role of This Bilinearity

Basis of All Tensor Manipulation Rules

Every standard algebraic manipulation performed on simple tensors, such as combining like terms, factoring scalars, and simplifying sums, ultimately traces back to repeated applications of the canonical map's bilinearity conditions rather than to any additional axioms specific to tensors.

Prototype for Multilinear Generalization

The two-argument bilinearity described here is the base case of the more general multilinearity required of the canonical map V₁ × V₂ × ... × Vₙ → V₁ ⊗ V₂ ⊗ ... ⊗ Vₙ, where additivity and scalar compatibility must hold independently in each of the n arguments, following the identical pattern established for two factors.