5.4.2 Tensor Product Space Addition Structure

Tensor Product Space Addition Structure is the specific vector space axiom governing how two elements of V ⊗ W are added, covering the definition of this addition on cosets, its interaction with decomposable elements, and the sense in which addition — unlike the scalar action — is what makes non-decomposable elements arise at all.

Definition of the Addition

Addition on Cosets

For elements x + R and y + R of V ⊗ W, addition is defined by

and this is well-defined because R is closed under addition: replacing x by x + r_1 and y by y + r_2 for r_1, r_2 in R changes x + y only by the element r_1 + r_2, which remains in R, leaving the resulting coset unchanged.

Addition Does Not Reduce to Addition Inside V or W

Adding v_1 ⊗ w and v_2 ⊗ w produces (v_1 + v_2) ⊗ w by the additivity relations imposed on R, so addition of two decomposable elements sharing a common factor collapses back to a single decomposable element; but adding v_1 ⊗ w_1 and v_2 ⊗ w_2 with no shared factor produces, in general, a genuinely non-decomposable element, since no relation in R identifies this sum with any single v ⊗ w.

The Axioms Satisfied

Abelian Group Structure

Addition on V ⊗ W is commutative and associative, has the coset 0 + R as identity, and every coset x + R has the additive inverse (−x) + R; all four facts are inherited directly from the corresponding facts already holding in the free vector space F(V × W) before the quotient, since a linear identity true before quotienting remains true for the corresponding cosets afterward.

Distributivity Linking Addition to the Scalar Action

Addition interacts with the scalar action described separately through the distributive law c(t + s) = ct + cs, guaranteeing that V ⊗ W under both its addition and its scalar action together satisfies the full set of vector space axioms, not merely the group axioms for addition in isolation.

Addition as the Source of Non-Decomposability

Where Genuinely New Elements Arise

Because every element of V ⊗ W is, by construction, a finite sum of decomposable elements, and the addition operation is exactly what combines these decomposable pieces into a single element of the space, addition is the mechanism responsible for the existence of non-decomposable elements such as e_1 ⊗ f_1 + e_2 ⊗ f_2. Neither the scalar action alone nor the decomposable elements alone can produce such an element; it is specifically the addition of two decomposable elements with no common factor that does.

Consequence for Tensor Rank

Because addition can combine decomposable elements without collapsing them into a single term, the tensor rank of a sum t_1 + t_2 is at most the sum of the ranks of t_1 and t_2, but can be strictly smaller if cancellation or recombination occurs; addition structure is therefore the operation that must be analyzed, term by term, whenever the minimal-length representation of a general element is sought.

Interaction with Coordinate Representation

Coefficientwise Addition in the Induced Basis

Relative to the induced basis {e_i ⊗ f_j}, adding two elements ∑ c_{ij}(e_i ⊗ f_j) and ∑ d_{ij}(e_i ⊗ f_j) produces ∑ (c_{ij} + d_{ij})(e_i ⊗ f_j), matching ordinary entrywise addition of coordinate arrays; this coordinatewise description is the concrete computational form of the addition defined abstractly on cosets above.