5.5.2 Tensor Elementary Factor Order

Tensor Elementary Factor Order is the significance of the position in which the two components of an elementary tensor v ⊗ w appear, establishing that v ⊗ w and w ⊗ v are, in general, elements of two different spaces V ⊗ W and W ⊗ V rather than two notations for a single underlying element, and examining the precise sense in which those two spaces are nonetheless related.

Order as Part of the Definition

The Left Position Is Reserved for V, the Right for W

By the construction of V ⊗ W, the canonical bilinear map has signature ⊗: V × W → V ⊗ W, built from the Cartesian product V × W taken in that specific order; the free vector space F(V × W) underlying the construction has one basis symbol for each pair (v, w) with v first and w second, and swapping the order would build a formal symbol (w, v) belonging to the different free vector space F(W × V).

w ⊗ v Belongs to a Different Space Unless V = W

If V and W are distinct spaces, w ⊗ v for w ∈ W and v ∈ V is not even an expression that parses as an element of V ⊗ W, since the roles of the two arguments are fixed by which space each factor is drawn from; w ⊗ v denotes an element of W ⊗ V instead, a space built from the same two factors but in the reverse order.

The Relationship Between V ⊗ W and W ⊗ V

A Canonical Isomorphism, Not an Equality

Although V ⊗ W and W ⊗ V are generally distinct as constructed objects, there is a canonical isomorphism

sending v ⊗ w to w ⊗ v and extended linearly; this map is well-defined by the universal property applied to the bilinear map (v, w) ↦ w ⊗ v, and it is an isomorphism because its inverse is built the same way from (w, v) ↦ v ⊗ w. The existence of τ justifies the informal statement that "the tensor product is commutative up to canonical isomorphism," while the order itself remains meaningful at the level of the underlying construction.

The Isomorphism Is Not the Identity When V = W

Even in the special case V = W, so that V ⊗ W and W ⊗ V are literally the same space, τ is generally not the identity map: τ(v_1 ⊗ v_2) = v_2 ⊗ v_1, which differs from v_1 ⊗ v_2 whenever v_1 ⊗ v_2 is not itself symmetric under the swap. This distinguishes elementary tensor order, which persists as a meaningful structural feature even when both factor spaces coincide, from a mere labeling convenience that would vanish once V = W.

Consequences for Iterated Tensor Products

Order Propagates to Longer Products

In an iterated tensor product V_1 ⊗ V_2 ⊗ ⋯ ⊗ V_k, the order of the factors fixes which position in the elementary tensor v_1 ⊗ v_2 ⊗ ⋯ ⊗ v_k corresponds to which space V_i; permuting the factor spaces produces a different, though canonically isomorphic, iterated tensor product, exactly as in the two-factor case, and this is what underlies the later distinction between covariant and contravariant slots in a general tensor of type (p, q), where swapping a V slot with a V* slot changes the type of tensor being described.