5.20.3 Tensor Product Linear Map Codomain Relation

Tensor Product Linear Map Codomain Relation is the fact that the codomain of the induced map f ⊗ g : V ⊗ W → V′ ⊗ W′ is exactly the tensor product V′ ⊗ W′ of the codomains of the individual maps f : V → V′ and g : W → W′, so that the tensor product operation on linear maps automatically pairs the tensor product of the domains with the tensor product of the codomains rather than mixing them arbitrarily. This relation ensures the tensor product of linear maps operation is internally consistent, always landing in the "correct" tensor product space, and it governs how codomains transform under composition, direct sums, and duality.

Statement of the Relation

The Basic Codomain Correspondence

Given f : V → V′ and g : W → W′, the induced linear map has signature

The codomain V′ ⊗ W′ is not an independent choice; it is forced to be the tensor product of the two individual codomains V′ and W′, since f ⊗ g is defined by sending v ⊗ w to f(v) ⊗ g(w), an element that by construction lies in V′ ⊗ W′.

Consistency with the Universal Property

This codomain assignment is exactly what the universal property delivers: the bilinear map (v, w) ↦ f(v) ⊗ g(w) has codomain V′ ⊗ W′ because f(v) ∈ V′ and g(w) ∈ W′, and the universal property of V ⊗ W produces the unique linear map into that same codomain, V′ ⊗ W′, factoring the bilinear map.

Codomain Behavior Under Special Cases

Identity on One Factor

When g = id_W, the codomain relation specializes to f ⊗ id_W : V ⊗ W → V′ ⊗ W, showing the codomain retains W unchanged in the factor where the identity is applied, while only the transformed factor's codomain changes from V to V′.

Composable Maps and Codomain Chaining

If f′ : V′ → V″ is a further linear map, the codomain relation composes coherently:

with the final codomain V″ ⊗ W′ correctly reflecting the composed codomain V″ of the first factor and the codomain W′ of the second, matching the composition identity (f′ ∘ f) ⊗ g = (f′ ⊗ id) ∘ (f ⊗ g).

Diagram of the Codomain Correspondence

Dimension Consequence of the Codomain Relation

Codomain Dimension Multiplies

For finite-dimensional spaces, the dimension of the codomain of f ⊗ g is dim(V′) · dim(W′), matching the general dimension relation for tensor product spaces applied specifically to the codomain, distinct from and generally unrelated to the rank of f ⊗ g itself, which is rank(f) · rank(g).

Codomain versus Image

The codomain V′ ⊗ W′ should not be conflated with the image of f ⊗ g, which is generally a proper subspace of V′ ⊗ W′ unless both f and g are surjective; the codomain relation fixes only the ambient space into which f ⊗ g maps, not the subspace it actually reaches.

Interaction with Duality

Codomain Relation for Dual Maps

If f : V → V′ has transpose (dual) map f* : V′* → V*, the codomain relation for the tensor product of transposes reads f* ⊗ g* : V′* ⊗ W′* → V* ⊗ W*, reversing the roles of domain and codomain spaces consistently with how duality reverses the direction of individual linear maps, while still respecting the same pattern of pairing tensor product domains with tensor product codomains.

Significance of the Codomain Relation

Guaranteeing Type-Correctness of Tensor Constructions

The codomain relation guarantees that compound constructions built from tensor products of linear maps remain type-correct at every stage, since the codomain of any tensor product map is always predictable in advance from the codomains of its constituent maps, without needing to inspect the map's action on individual elements.

Foundation for Functorial Reasoning

Because the codomain of f ⊗ g is determined functorially by the codomains of f and g, the codomain relation is a basic ingredient in treating ⊗ as a bifunctor on the category of vector spaces, supporting systematic reasoning about how transformations and their codomains propagate through tensor constructions of arbitrary complexity.