5.20 Tensor Product of Linear Maps Operation

Tensor Product of Linear Maps Operation is the construction that takes two linear maps f : V → V′ and g : W → W′ and produces a single linear map f ⊗ g : V ⊗ W → V′ ⊗ W′, defined on simple tensors by (f ⊗ g)(v ⊗ w) = f(v) ⊗ g(w) and extended by linearity, so that linear transformations on the factor spaces combine into a linear transformation on the tensor product itself. This operation makes the tensor product functorial, turning pairs of linear maps into a single map compatible with composition and identities, and it is the mechanism by which operators, symmetries, and transformations propagate from vector spaces to their tensor products.

Definition of the Induced Map

Action on Simple Tensors

Given linear maps f : V → V′ and g : W → W′, the tensor product map f ⊗ g is defined on simple tensors by

Extension by Linearity to General Tensors

Since every element of V ⊗ W is a finite sum Σ vₖ ⊗ wₖ, the map is extended by declaring

Well-Definedness via the Universal Property

That this rule produces a genuine, well-defined linear map (rather than depending on how a given element happens to be written as a sum of simple tensors) is not automatic and must be justified: the bilinear map (v, w) ↦ f(v) ⊗ g(w) from V × W to V′ ⊗ W′ induces, by the universal property of the tensor product, a unique linear map V ⊗ W → V′ ⊗ W′ agreeing with f ⊗ g on simple tensors, which guarantees consistency regardless of the representation chosen.

Functorial Properties

Compatibility with Composition

If f : V → V′, f′ : V′ → V″, g : W → W′, and g′ : W′ → W″ are linear maps, the tensor product operation satisfies

so that tensoring maps commutes with composing them.

Compatibility with Identities

For identity maps, id_V ⊗ id_W = id_{V ⊗ W}, confirming that the tensor product operation on maps preserves the identity transformation, which together with the composition rule shows the assignment (V, W) ↦ V ⊗ W, (f, g) ↦ f ⊗ g is a bifunctor on the category of vector spaces.

Matrix Representation: The Kronecker Product

Component Description in Finite Dimensions

When V, V′, W, W′ are finite-dimensional with chosen bases, and f, g are represented by matrices A (size m′ × m) and B (size n′ × n) respectively, the map f ⊗ g is represented, in the induced basis {eᵢ ⊗ fⱼ}, by the Kronecker product matrix A ⊗ B, whose block form is

giving the abstract operation f ⊗ g a completely explicit, computable matrix form widely used in numerical linear algebra.

Diagram of the Induced Map

Preservation of Structural Properties

Injectivity and Surjectivity

If f and g are both injective, f ⊗ g is injective; if both are surjective, f ⊗ g is surjective. In particular, if f and g are both isomorphisms, f ⊗ g is an isomorphism with inverse f⁻¹ ⊗ g⁻¹, so the tensor product of linear maps respects invertibility.

Rank Behavior

The rank of f ⊗ g equals the product of the ranks of f and g, mirroring the multiplicative dimension relation of the tensor product itself and following directly from the Kronecker product matrix representation, since the rank of a Kronecker product of matrices is the product of the individual ranks.

Significance of the Operation

Propagating Transformations to Tensor Product Spaces

The tensor product of linear maps operation is the standard way to lift transformations defined on individual vector spaces to transformations on their tensor products, which is essential whenever a system is built by combining subsystems, such as forming the joint evolution operator of a composite quantum system from the operators on its parts, or applying independent transformations to the two indices of a matrix-like tensor.

Foundation for Functorial and Categorical Treatments

Because the operation respects composition and identities, it establishes the tensor product as a bifunctor, providing the categorical foundation used to study naturality, symmetric monoidal structure, and coherence in the broader theory of tensor categories built on top of tensor product vector spaces.