5.4.3 Tensor Product Space Scalar Action Structure

Tensor Product Space Scalar Action Structure is the specific vector space axiom governing how the base field F acts on V ⊗ W by scalar multiplication, covering the definition of this action on cosets, its interaction with decomposable elements, and its role among the vector space axioms that make V ⊗ W a vector space over F.

Definition of the Action

Scalar Multiplication on Cosets

Writing an element of V ⊗ W as a coset x + R of the free vector space F(V × W) modulo the relation subspace R, the scalar action is defined by

for c in F, which is well-defined because R is closed under scalar multiplication: if x_1 + R = x_2 + R, then x_1 − x_2 is in R, and c(x_1 − x_2) = cx_1 − cx_2 remains in R, so cx_1 + R = cx_2 + R regardless of which representative was scaled.

Passing the Scalar Through Either Factor

On decomposable elements, this action satisfies

with all three expressions denoting the same element of V ⊗ W; this three-way equality is not an accident of computation but a direct restatement of the homogeneity generators imposed on R during bilinear relation imposition, which forced exactly these cosets to coincide.

The Axioms Satisfied

The Field Axioms Carried Over from the Quotient

The scalar action on V ⊗ W satisfies the usual vector space axioms relating scalar multiplication to addition and to the field's own operations —

for t, s in V ⊗ W and c, d in F; each of these holds because the corresponding identity already holds coordinatewise in the free vector space F(V × W) before the quotient, and passing to cosets preserves any identity that held before quotienting.

Not an Independent Structure to Verify from Scratch

Because V ⊗ W is defined as a quotient vector space, the scalar action inherits every one of these axioms automatically from the general theory of quotient vector spaces; no separate, tensor-product-specific proof of the axioms is required beyond confirming the well-definedness of the action on cosets.

Interaction with Other Structural Facts

Basis Coordinates Scale Coordinatewise

Relative to the induced basis {e_i ⊗ f_j}, scaling an element ∑_{i,j} c_{ij}(e_i ⊗ f_j) by a field element c scales every coefficient c_{ij} by c, matching ordinary scalar multiplication of a coordinate array; this is the concrete, coordinate-level manifestation of the abstractly defined scalar action described above.

Distinguishing Scalar Action from the Tensor Operation Itself

The scalar action c · t should not be confused with tensoring by a scalar treated as a 1 × 1 space; scalars act on V ⊗ W externally, as coefficients, while ⊗ combines two vectors internally to produce a new element, and the equality c(v ⊗ w) = (cv) ⊗ w is precisely the identity that reconciles these two otherwise distinct operations.