5.2.1 Tensor Product Construction Area

The tensor product construction area explores how to build tensor spaces from vector spaces, defining bilinear mappings and foundational structures in multilinear algebra.

Tensor Product Construction Area is the detailed procedure by which the tensor product V ⊗ W of two vector spaces is produced as a concrete quotient of a free vector space, covering the formal sums used as a starting point, the specific relations imposed to force bilinearity, and the resulting identification of v ⊗ w as the image of a pair (v, w) under the quotient map.

Starting from Formal Sums

The Free Vector Space on Pairs

Given vector spaces V and W over a field F, let F(V × W) denote the free vector space with basis the set V × W itself — one formal basis vector for every pair (v, w), with no relation yet connecting these basis vectors to the vector space structure already present on V and W. A general element of F(V × W) is a finite formal sum

\sum_{i = 1}^{n} c_{i} (v_{i}, w_{i})

with scalars c_i and no simplification available between distinct terms, since (v_1, w) and (v_2, w) are unrelated basis vectors even if v_1 and v_2 are related in V.

Imposing the Relations

The Subspace of Relations

Let R be the subspace of F(V × W) generated by all elements of the four forms

(v_{1} + v_{2}, w) - (v_{1}, w) - (v_{2}, w),

(v, w_{1} + w_{2}) - (v, w_{1}) - (v, w_{2}),

(c v, w) - c (v, w),

(v, c w) - c (v, w)

for all v, v_1, v_2 in V, w, w_1, w_2 in W, and c in F. The tensor product is defined as the quotient

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

and the image of (v, w) under the quotient map is written v ⊗ w.

Why Exactly These Four Relations

Each of the four generating relations corresponds to exactly one of the two bilinearity conditions applied to one of the two arguments: additivity in the first argument, additivity in the second, homogeneity in the first, and homogeneity in the second. No relation is included that mixes v_1, v_2 with w_1, w_2 simultaneously, since bilinearity is checked one argument at a time, not jointly, matching the multilinear map boundary's insistence on separate rather than joint linearity.

Consequences of the Quotient

The Canonical Map Is Bilinear by Construction

The map ⊗: V × W → V ⊗ W sending (v, w) to v ⊗ w is bilinear precisely because the relations quotiented out are exactly the failures of bilinearity present in F(V × W); each generating relation of R becomes the zero element in the quotient, forcing the corresponding bilinearity identity to hold for ⊗.

Every Element Is a Finite Sum of Decomposable Elements

Since F(V × W) is spanned by the basis vectors (v, w), and the quotient map is linear and surjective, every element of V ⊗ W is a finite sum ∑ c_i (v_i ⊗ w_i), and absorbing each scalar c_i into either factor gives a finite sum of decomposable elements v_i ⊗ w_i with no scalar coefficients remaining outside the tensor symbol.