5.20.4 Tensor Product Linear Map Elementary Action

The tensor product linear map elementary action combines vectors and tensors to produce new linear transformations through bilinear operations.

Tensor Product Linear Map Elementary Action is the single defining rule (f ⊗ g)(v ⊗ w) = f(v) ⊗ g(w), prescribing how the tensor product map f ⊗ g acts on simple tensors, from which the entire behavior of f ⊗ g on all of V ⊗ W is generated by linear extension. This elementary action is the minimal piece of data needed to specify f ⊗ g completely, and every other property of the tensor product of linear maps — well-definedness, functoriality, matrix representation, rank — is derived from this one rule together with the universal property that justifies extending it consistently to general tensors.

The Elementary Rule

Statement on Simple Tensors

For linear maps f : V → V′ and g : W → W′, the elementary action specifies

(f \otimes g) (v \otimes w) = f (v) \otimes g (w)

for every v ∈ V and w ∈ W. Nothing else is specified directly; the action on any element that is not a simple tensor is determined only indirectly, via linearity.

Why a Rule on Simple Tensors Suffices

Because simple tensors v ⊗ w span V ⊗ W, a linear map is completely determined once its values on all simple tensors are known, provided those values are consistent with the bilinearity that simple tensors already satisfy — namely (v₁ + v₂) ⊗ w = v₁ ⊗ w + v₂ ⊗ w and (av) ⊗ w = a(v ⊗ w), and similarly in the second slot. The elementary action respects these relations automatically, since f(v₁ + v₂) ⊗ g(w) = f(v₁) ⊗ g(w) + f(v₂) ⊗ g(w) by linearity of f and bilinearity of ⊗.

From Elementary Action to General Action

Extending by Linearity

For a general element t = Σ vₖ ⊗ wₖ ∈ V ⊗ W, the elementary action forces

(f \otimes g) (t) = \sum_{k} f (v_{k}) \otimes g (w_{k})

so the entire map is nothing more than repeated application of the one elementary rule to each simple-tensor summand.

The Well-Definedness Issue the Elementary Action Alone Cannot Resolve

A subtlety arises because a given tensor t can be written as a sum of simple tensors in more than one way, so the formula above must give the same answer for every representation of t. The elementary action by itself does not prove this; it is guaranteed instead by the universal property, which shows the bilinear map (v, w) ↦ f(v) ⊗ g(w) factors through a genuine, single-valued linear map on V ⊗ W, independent of how any particular element is decomposed into simple tensors.

Diagram of the Elementary Action

Instances of the Elementary Action

Identity Maps

Taking f = id_V and g = id_W, the elementary action reduces to (id_V ⊗ id_W)(v ⊗ w) = v ⊗ w, confirming id_V ⊗ id_W = id_{V ⊗ W} at the level of simple tensors, and hence, by linear extension, on the whole space.

Zero Map

If f = 0, the elementary action gives (0 ⊗ g)(v ⊗ w) = 0 ⊗ g(w) = 0, since any simple tensor with a zero factor is the zero element of the tensor product, showing the elementary action correctly forces 0 ⊗ g to be the zero map regardless of g.

Scalar Multiples

For a scalar a, (af) ⊗ g acts by (af)(v) ⊗ g(w) = a(f(v) ⊗ g(w)), showing the elementary action is itself bilinear in the pair (f, g), matching the fact that ⊗ on the space of linear maps Hom(V,V′) ⊗ Hom(W,W′) → Hom(V ⊗ W, V′ ⊗ W′) is a linear (in fact injective, in finite dimensions bijective) map.

Elementary Action in Coordinates

Matrix Entries from the Elementary Action

In finite dimensions, applying the elementary action to basis simple tensors eᵢ ⊗ fⱼ gives (f ⊗ g)(eᵢ ⊗ fⱼ) = f(eᵢ) ⊗ g(fⱼ), and expanding f(eᵢ) and g(fⱼ) in the target bases directly produces the entries of the Kronecker product matrix A ⊗ B representing f ⊗ g, so the elementary action is the computational source of every entry in the matrix representation.

Significance of the Elementary Action

The Minimal Specification Principle

The elementary action embodies a general principle in tensor product theory: any linear map out of a tensor product, including f ⊗ g itself, is completely and economically specified by its values on simple tensors, provided those values are compatible with bilinearity — a principle that recurs throughout the study of tensor products and their associated maps.

Basis for All Higher Properties

Every subsequent property of the tensor product of linear maps operation — composition compatibility, identity preservation, rank multiplicativity, Kronecker product representation — is a consequence that can be checked directly from the elementary action on simple tensors and then transferred to the whole space by linearity, making the elementary action the true computational and conceptual root of the entire operation.