5.11.1 Tensor Product First Factor Linearity

Tensor Product First Factor Linearity describes how the first factor in a tensor product behaves linearly, forming a foundational property in multilinear algebra.

Tensor Product First Factor Linearity is the specific instance of the tensor product's general multilinearity that isolates the first argument alone: with every other factor held fixed, the map sending a vector v1 in V1 to the tensor v1 ⊗ v2 ⊗ ⋯ ⊗ vn is linear, satisfying additivity and scalar compatibility purely with respect to that one slot.

Formal Statement

Fix vectors v2 in V2, …, vn in Vn. First factor linearity asserts that the map

L_{v_{2}, \dots, v_{n}} : V_{1} \to V_{1} \otimes \dots \otimes V_{n}, L_{v_{2}, \dots, v_{n}} (x) = x \otimes v_{2} \otimes \dots \otimes v_{n}

is linear: L(x + y) = L(x) + L(y) and L(λx) = λL(x) for all x, y in V1 and scalars λ. This is precisely one of the n separate linearity requirements bundled together in the general multilinearity of the tensor product, singled out here because the first slot often plays a distinguished role in specific computations and proofs.

Derivation from the General Multilinearity

First factor linearity is not an independent postulate but a direct specialization of the universal property's multilinearity requirement to the case where only the first argument is allowed to vary.

Restriction of the Universal Multilinear Map

The canonical multilinear map τ defining the tensor product satisfies, by definition, linearity in every one of its n arguments. Restricting τ to vary only its first argument, with the remaining n − 1 arguments frozen at v2, …, vn, produces exactly the map L above, and its linearity is inherited automatically from the linearity of τ in that slot.

Explicit Additivity and Scalar Identities

(x_{1} + x_{2}) \otimes v_{2} \otimes \dots \otimes v_{n} = x_{1} \otimes v_{2} \otimes \dots \otimes v_{n} + x_{2} \otimes v_{2} \otimes \dots \otimes v_{n}

together with (λx1) ⊗ v2 ⊗ ⋯ ⊗ vn = λ(x1 ⊗ v2 ⊗ ⋯ ⊗ vn), give the complete content of first factor linearity, holding for every fixed choice of v2, …, vn.

Use in Constructing Linear Maps by Currying

First factor linearity is the specific fact that permits the tensor product to be reinterpreted, one slot at a time, as an ordinary linear map with values in a space of linear maps on the remaining factors.

Currying the First Slot

Because L is linear in its own right for every fixed choice of v2, …, vn, one can construct a single linear map from V1 into the space of linear maps Hom(V2 ⊗ ⋯ ⊗ Vn, V1 ⊗ ⋯ ⊗ Vn), sending v1 to the linear map "tensor with v1 on the left." This currying maneuver, isolating exactly the first-factor linearity, underlies the standard identification of the tensor product with iterated Hom-spaces used throughout the study of multilinear maps.

Basis Expansion in the First Factor

First factor linearity is also what justifies expanding a tensor's first-factor contribution in a chosen basis of V1 independently of how the remaining factors are represented: writing v1 as a linear combination of basis vectors and distributing the tensor product across that sum, using only linearity in the first slot, produces a corresponding decomposition of v1 ⊗ v2 ⊗ ⋯ ⊗ vn as a sum over the basis vectors of V1 alone, with the remaining factors carried along unchanged in every term.

Role in Proofs by Slot-by-Slot Reduction

Because the tensor product is multilinear one slot at a time, many proofs about general multilinear identities proceed by fixing all but one slot, invoking first factor linearity (or its analogue in whichever slot is currently varying), and then repeating the argument slot by slot until all n factors have been addressed.

Reduction to the Single-Slot Case

A proof that a certain identity holds for all tensors in V1 ⊗ ⋯ ⊗ Vn can often be reduced, via first factor linearity, to checking the identity only on tensors of the form e ⊗ v2 ⊗ ⋯ ⊗ vn for e ranging over a basis of V1, since linearity in the first slot extends any identity verified there to all of V1 by linear combination, with the remaining slots subsequently handled the same way.

Illustrative Diagram

The dashed box marks the single slot allowed to vary while the remaining, solid-bordered factors stay fixed, isolating exactly the linearity captured by first factor linearity.