14.4.3 Tensor Linear Operator Product Codomain Space

The Tensor Linear Operator Product Codomain Space defines the output space of tensor operator products, specifying the range of resulting transformations.

Tensor Linear Operator Product Codomain Space is the tensor product V tensor W of the two spaces into which the individual operators f and g map their outputs, serving as the single target space of the induced operator f tensor g and, since the operator case forces domain and codomain to coincide, matching exactly the domain space of the same construction.

Formation of the Codomain Space

Built from the Same Spaces as the Domain

Given operators

f : V \to V

g : W \to W

the codomain space of the induced operator is

V \otimes W,

identical as a space to the domain, since the codomain of f is V and the codomain of g is W, exactly matching the spaces used to build the domain of f tensor g.

Why the Codomain Space Deserves Separate Attention

Even though the codomain space coincides with the domain space as a set, it plays a logically distinct role, receiving the outputs of the elementary action rather than supplying the inputs, and this distinction remains meaningful when tracking how elements move through the operator, particularly when the operator is applied repeatedly or composed with others.

Elements of the Codomain Space

Elementary Tensor Outputs

Every output of the elementary action, namely f of v tensor g of w, is itself an elementary tensor lying in the codomain space, built from the image of v under f and the image of w under g, so the codomain space receives exactly the same kind of generating elementary tensors that span the domain space.

General Elements as Images of Sums

Because the elementary action extends by linearity, a general element of the codomain space arising as the image of a sum of elementary tensors is itself a sum of elementary tensor outputs, so the codomain space, though equal as a set to the domain space, is reached by the operator through the same additive combination process used to build up its domain elements.

Coordinate Description of the Codomain Space

Kronecker Product Column Space

With bases fixed on V and W, the image of f tensor g, viewed inside the codomain space, corresponds to the column space of the Kronecker product matrix A tensor B, where A and B are the matrices of f and g, so questions about which elements of the codomain space are actually reached by the operator reduce to standard column-space computations on this explicit matrix.

Full Codomain versus Image

The codomain space itself is the entire space V tensor W, regardless of whether f tensor g is surjective, while the image is the possibly smaller subspace actually reached; the rank formula

rank (f \otimes g) = rank (f) \cdot rank (g)

measures how much of the codomain space is filled by the image, with equality to the dimension of the codomain space occurring exactly when both f and g are individually surjective.

The Codomain Space Under Composition

Serving as an Intermediate Space

When two operator products are composed, the codomain space of the first, namely V tensor W, must coincide with the domain space of the second for the composite to be defined; because domain and codomain always coincide in the operator case, this matching is automatic whenever the same pair of spaces V and W is used throughout the chain of composition.

Stability Under Repeated Composition

Because the codomain space equals the domain space, the operator f tensor g can be composed with itself any number of times, producing powers of the form (f tensor g) to the k, each such power remaining an operator on the same fixed codomain space V tensor W, and satisfying

{(f \otimes g)}^{k} = f^{k} \otimes g^{k}

by repeated application of the composition identity for tensor products of maps.