5.14.1 Tensor Canonical Map Domain Product

The Tensor Canonical Map Domain Product links tensor spaces through bilinear mappings, establishing foundational structures in multilinear algebra.

Tensor Canonical Map Domain Product is the Cartesian product V × W of two vector spaces, understood specifically in its role as the domain of the canonical tensor map ⊗: V × W → V ⊗ W. Rather than being studied as a vector space in its own right, this product is examined here for the particular structure it must carry in order to serve as the source of a well-defined universal bilinear map, distinguishing it from the tensor product's codomain and from the direct sum construction.

The Product as a Domain

Set-Theoretic Structure of the Domain

As a domain, V × W consists of all ordered pairs (v, w) with v ∈ V and w ∈ W. Considered purely as a set, this domain carries no algebraic structure beyond what is inherited componentwise from V and W.

Vector Space Structure Inherited Componentwise

When endowed with componentwise addition and scalar multiplication:

(v, w) + (v^{'}, w^{'}) = (v + v^{'}, w + w^{'})

c (v, w) = (c v, c w)

the domain V × W becomes the direct sum (or direct product, in the finite case) V ⊕ W, a vector space of dimension dim(V) + dim(W) in the finite-dimensional case.

Why This Vector Space Structure Is Not the One Used by the Tensor Map

The Domain's Linear Structure Is Irrelevant to Bilinearity

The tensor map ⊗: V × W → V ⊗ W is not linear with respect to the direct sum structure on V × W; it is bilinear. Bilinearity is a distinct and generally incompatible requirement from linearity on the product, since a linear map on V ⊕ W would need to satisfy (v, w) + (v', w') ↦ (v, w) ↦ ... + (v', w') ↦ ... in a way that does not match how v ⊗ w behaves under componentwise addition of pairs.

Distinguishing the Domain Product from the Tensor Product

This distinction is the reason the notation V × W (or sometimes V ⊕ W) is kept strictly separate from V ⊗ W: the former is the domain on which bilinear maps are defined, of dimension dim(V) + dim(W), while the latter is the target of the universal bilinear map, of dimension dim(V) · dim(W).

Dimensional Comparison

Additive Dimension of the Domain

For finite-dimensional V and W:

dim (V \times W) = dim (V) + dim (W)

Multiplicative Dimension of the Tensor Product

dim (V \otimes W) = dim (V) \cdot dim (W)

This dimensional contrast is a direct consequence of the domain being used only as an indexing set for bilinear inputs, rather than as the vector space over which linearity is imposed.

The Domain in the Free Module Construction

Basis Elements Indexed by the Domain

In the free-module construction of the tensor product, the domain V × W supplies the generating set: the free module F(V × W) has one basis element for every pair (v, w), regardless of any linear relation these pairs might satisfy inside V ⊕ W.

Quotienting Away the Product's Own Linear Structure

The bilinear relations subsequently imposed on F(V × W) do not correspond to the linear relations already present in V ⊕ W; rather, they encode an entirely separate structure, namely bilinearity, which must be added on top of the raw indexing set before the quotient can be identified with V ⊗ W.

Visualizing the Domain

Practical Consequences

Choosing Domain Bases for Bilinear Forms

When representing a bilinear map by a matrix, the choice of bases for V and W separately, rather than a single combined basis for V × W as a direct sum, reflects the fact that the domain is being used positionally, as a pair of independent inputs, not as a single linear argument.

Contrast with Multilinear Domains

The same principle extends to multilinear maps of n arguments, where the domain V₁ × V₂ × ... × Vₙ again serves purely as an indexing structure for the inputs, while the target of the universal multilinear map, the iterated tensor product, carries the multiplicative dimension dim(V₁) · dim(V₂) ··· dim(Vₙ).