1.8.3 Linear Map Tensor Abstraction

Linear Map Tensor Abstraction formalizes how linear transformations act on tensor spaces, bridging algebraic structures with geometric interpretations.

Linear Map Tensor Abstraction is the extension of tensor abstraction from vector spaces themselves to the linear maps between them, treating the tensor product not merely as an operation producing a new space from two given spaces but as an operation producing a new linear map from two given linear maps, so that the tensor product becomes a construction acting simultaneously and compatibly on both objects and the morphisms connecting them. It supplies the precise sense in which tensor product "respects" linear maps, which is the abstract ingredient that makes pushing tensors forward along linear transformations, and pulling structures back along them, into well-defined operations.

The Tensor Product of Two Linear Maps

Constructing a New Map from Two Old Ones

Given linear maps f : V → V′ and g : W → W′, the tensor product construction produces a single linear map f ⊗ g from V ⊗ W to V′ ⊗ W′, defined on simple tensors by applying f and g separately to each factor and extended linearly to the whole space.

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

Well-Definedness from the Universal Property

That this rule extends consistently to a genuine linear map, rather than merely a rule on individual simple tensors, is guaranteed by the universal property of the tensor product: the assignment (v, w) ↦ f(v) ⊗ g(w) is bilinear in v and w, so it factors uniquely through V ⊗ W, producing f ⊗ g automatically.

Functoriality of the Tensor Product

Respecting Composition

The tensor product of linear maps respects composition, meaning that tensoring two composed maps agrees with composing the two tensored maps, which is the precise functorial statement underlying linear map tensor abstraction.

(f^{'} \circ f) \otimes (g^{'} \circ g) = (f^{'} \otimes g^{'}) \circ (f \otimes g)

Respecting Identities

Tensoring two identity maps produces the identity map on the tensor product space, the second half of the functoriality requirement, confirming that the tensor product construction on linear maps behaves consistently with the identity morphisms of the underlying category of vector spaces.

{id}_{V} \otimes {id}_{W} = {id}_{V \otimes W}

Pushing Tensors Forward Along a Linear Map

Transporting a Tensor to a New Space

Given a linear map f : V → V′, the induced map f^{⊗ p} acting on V^{⊗ p} allows a type (p, 0) tensor built from V to be pushed forward to a corresponding tensor built from V′, extending the action of f on individual vectors to an action on tensors of every rank built purely from upper indices.

f^{\otimes p} (T) \in V^{' \otimes p}

Pulling Covariant Tensors Backward

Because a linear map f : V → V′ induces a dual map f^* : V′^* → V^* running in the opposite direction, tensors built purely from lower indices, type (0, q) tensors, are naturally pulled back rather than pushed forward, following the contravariant behavior of the dual construction applied at the level of maps.

f^{*} : {(V^{'})}^{*} \to V^{*}

Mixed-Type Tensors and Restricted Applicability

Why Only Isomorphisms Transport Mixed Tensors Fully

For a tensor of mixed type (p, q) with q > 0, transporting it fully along a linear map f that is not invertible runs into the obstruction that the lower indices require a map in the direction opposite to f, so a fully general transport of a type (1, 1) tensor, for instance, is only automatically available when f is an isomorphism, allowing f^{-1} to substitute for the missing dual direction on the lower index.

f \otimes {(f^{- 1})}^{*} : V \otimes V^{*} \to V^{'} \otimes {(V^{'})}^{*}

Consistency with the Classical Change-of-Basis Case

Restricting this discussion to the case where f is a change-of-basis isomorphism recovers exactly the classical tensor transformation law, confirming that linear map tensor abstraction is the general categorical statement of which the ordinary component transformation rule is one specific instance, applied when f happens to relate two bases of the same space.

The Hom-Tensor Relationship as a Functorial Statement

Naturality of the Isomorphism

The isomorphism between Hom(V, W) and V^* ⊗ W, available for finite-dimensional vector spaces, is not merely an isomorphism for each fixed pair of spaces but a natural one, meaning it is compatible with the induced action of linear maps on both sides, which is the precise sense in which this isomorphism itself qualifies as tensorial.

Diagrammatic Summary

The diagram shows the functorial content of linear map tensor abstraction: the pair of maps f and g acting on V and W induces a single well-defined map f ⊗ g acting on V ⊗ W, consistent with composition and identities.