5.3.3 Tensor Product Bilinear Relation Imposition

Tensor Product Bilinear Relation Imposition is the step of the tensor product construction in which the subspace R of the free vector space F(V × W) is defined by choosing generators that encode exactly the additivity and homogeneity conditions a bilinear map must satisfy in each argument separately, fixing what quotient formation will later collapse to zero.

Choosing the Generators

Four Families, One per Bilinearity Condition

The subspace R is generated by all elements of the four forms

ranging over all v, v_1, v_2 in V, w, w_1, w_2 in W, and c in F. The first two generate additivity in the first and second argument respectively; the last two generate homogeneity in the first and second argument respectively.

R Is the Span, Not Just the Listed Generators

R consists of every finite linear combination of these generator elements, not merely the generators themselves; this matters because quotient formation requires a full subspace, closed under addition and scalar multiplication, and the four families above are a generating set for that subspace rather than an exhaustive listing of its elements.

What Each Family Encodes

Additivity Generators Force Distributivity Over Sums

Quotienting by the first family forces (v_1 + v_2, w) and (v_1, w) + (v_2, w) to become equal cosets, since their difference is precisely a generator of R; this is what will make the canonical map distribute over addition in its first argument once the quotient is taken.

Homogeneity Generators Force Scalars to Pull Through

Quotienting by the third and fourth families forces (cv, w) to become equal to c(v, w) as a coset, and similarly in the second argument; this is what allows a scalar to be moved freely between the two tensor factors, so that (cv) ⊗ w = v ⊗ (cw) = c(v ⊗ w) once the quotient map is applied.

Why No Cross-Term Relation Is Included

Bilinearity Is Argument-by-Argument, Not Joint

No generator of R involves varying both v and w simultaneously, such as a relation between (v_1 + v_2, w_1 + w_2) and a four-term expansion; including such a relation would impose joint linearity, which is a strictly stronger and different condition than bilinearity. The absence of any such generator is what preserves the characteristic non-joint-linear behavior described in the bilinear map boundary, where B(u_1 + u_2, v_1 + v_2) expands into four separate cross terms rather than collapsing to two.

Consequences for the Next Stage

R Fully Determines the Quotient to Follow

Once the generators of R are fixed, quotient formation proceeds mechanically: every fact about the resulting space V ⊗ W, including the bilinearity of the canonical map ⊗, traces back to exactly these four families of relations and no others, making bilinear relation imposition the step that fully determines what algebraic identities will hold in the tensor product before the quotient itself is even formed.