5.11.2 Tensor Product Second Factor Linearity

Tensor Product Second Factor Linearity explains how the second factor behaves under linear maps, key to multilinear algebra.

Tensor Product Second Factor Linearity is the counterpart to first factor linearity: with the first (and any further) factor held fixed, the map sending a vector v2 in V2 to the tensor v1 ⊗ v2 ⊗ v3 ⊗ ⋯ ⊗ vn is linear, satisfying additivity and scalar compatibility purely with respect to the second slot, independently of how the tensor product behaves in any other position.

Formal Statement

Fix v1 in V1, v3 in V3, …, vn in Vn. Second factor linearity asserts that the map

M_{v_{1}, v_{3}, \dots, v_{n}} : V_{2} \to V_{1} \otimes \dots \otimes V_{n}, M_{v_{1}, v_{3}, \dots, v_{n}} (y) = v_{1} \otimes y \otimes v_{3} \otimes \dots \otimes v_{n}

is linear: M(y1 + y2) = M(y1) + M(y2) and M(λy) = λM(y) for all y1, y2 in V2 and scalars λ. Together with first factor linearity and its analogues in every remaining slot, this constitutes one of the n separate linearity conditions that jointly define the tensor product's multilinearity.

Symmetry Between First and Second Factor Linearity

Although first and second factor linearity are stated with respect to different slots, they are not independent facts requiring separate justification in the two-factor case; they are related by the same underlying symmetry that governs the tensor product's construction.

The Swap Isomorphism

For two factors, there is a canonical isomorphism V1 ⊗ V2 ≅ V2 ⊗ V1 sending v ⊗ w to w ⊗ v. Under this isomorphism, second factor linearity of the tensor product on V1 ⊗ V2 corresponds exactly to first factor linearity of the tensor product on V2 ⊗ V1, so the two properties are, up to this canonical relabeling, the same statement viewed from opposite sides.

Independent Verification Still Required

Despite this symmetry, second factor linearity must still be explicitly verified (or built into the construction) for the tensor product on V1 ⊗ V2 as originally presented, since the swap isomorphism is a separate structural fact about the tensor product that itself requires the multilinearity of τ to construct in the first place; the symmetry clarifies the relationship between the two linearity properties without eliminating the need to establish both.

Role in Bilinear Form Theory

Second factor linearity carries particular weight in the classical theory of bilinear forms, where it appears as the requirement that a form be linear in its second argument.

Bilinear Forms as Tensor Product Duals

A bilinear form B on V1 × V2 corresponds, via the universal property, to a linear functional on V1 ⊗ V2. Linearity of B in its second argument — the direct analogue of second factor linearity — is exactly what allows B to be reconstructed from its values on pairs of basis vectors by the same distribution-over-sums argument used generally for multilinear maps, and it is one of the two conditions (together with first-argument linearity) that qualifies B as bilinear in the classical sense.

Adjoint Operators and the Second Slot

When V2 carries additional structure, such as an inner product, second factor linearity is what permits the definition of an adjoint operator with respect to that slot: fixing the first argument and varying the second linearly allows the resulting linear functional on V2 to be represented, via the Riesz representation or its finite-dimensional analogue, by a unique vector, giving rise to operators built directly from second-slot linear dependence.

Use in Slot-by-Slot Proof Strategies

Just as first factor linearity permits reduction of an identity to basis vectors of V1 with the remaining slots fixed, second factor linearity permits the analogous reduction with the roles of the first and second slots exchanged.

Iterated Reduction Across Two Slots

A proof establishing a bilinear identity for all of V1 × V2 typically proceeds by fixing an arbitrary element of V2, reducing to basis vectors of V1 using first factor linearity, and then, for each resulting case, invoking second factor linearity to reduce the remaining freedom in V2 to its own basis vectors — verifying the identity in the end only on the finitely many pairs of basis vectors, then extending by the two linearity properties in sequence.

Illustrative Diagram

The dashed box marks the second slot, the one allowed to vary under second factor linearity, while the first slot, drawn with a solid border, remains fixed throughout.