2.22.3 Tensor Linear Map Additivity Preservation

Tensor Linear Map Additivity Preservation is the property that the tensor identification of linear maps as elements of V* ⊗ W respects addition on both sides of the correspondence, so that the tensor representing a sum of two linear maps equals the sum of the tensors representing each map individually, and conversely, adding two tensors in V* ⊗ W and reinterpreting the result as a linear map produces exactly the pointwise sum of the two original maps. This preservation of additive structure is what allows Hom(V, W) and V* ⊗ W to be treated as literally the same vector space rather than merely as two spaces that happen to be in bijection.

Additivity on the Side of Linear Maps

Pointwise Sum Definition

For φ, ψ ∈ Hom(V, W), the sum φ + ψ is defined pointwise by:

for every v ∈ V. This is the standard vector-space addition on Hom(V, W), applied to functions in the usual pointwise manner.

Additivity Under the Correspondence

Additivity preservation states that if φ corresponds to the tensor α ∈ V* ⊗ W and ψ corresponds to the tensor β ∈ V* ⊗ W, then φ + ψ corresponds to α + β, the ordinary sum of tensors in V* ⊗ W. No correction term or additional operation is needed; the correspondence Hom(V, W) ≅ V* ⊗ W is not merely a bijection of sets, but an isomorphism of vector spaces, and additivity preservation is precisely the additive half of that isomorphism property.

Verification in Coordinates

Matrix Addition Matches Tensor Addition

Relative to bases {e_i} of V and {f_j} of W, if φ has matrix entries a^j_i and ψ has matrix entries b^j_i, then:

so the matrix of φ + ψ has entries a^j_i + b^j_i, exactly matching the entrywise sum of the two tensors Σ a^j_i e^i ⊗ f_j and Σ b^j_i e^i ⊗ f_j when they are added as elements of V* ⊗ W. Matrix addition, entry by entry, is the coordinate expression of tensor addition applied to the (1, 1)-tensor representatives of linear maps.

Additivity as Linearity of the Identification Map

The Identification Map Itself Is Linear

Denote by Θ : V* ⊗ W → Hom(V, W) the map sending a simple tensor ω ⊗ w to the rank-one linear map v ↦ ω(v) w, extended linearly to all of V* ⊗ W. Additivity preservation is equivalent to the statement that Θ is a linear map, since Θ(α + β) = Θ(α) + Θ(β) is exactly additivity, and Θ(cα) = cΘ(α) is the corresponding statement for scalar multiples. Because Θ is defined by linear extension from simple tensors in the first place, its linearity, and hence additivity preservation, is guaranteed by construction rather than requiring separate proof.

Role of Bilinearity of the Tensor Product

The universal property of the tensor product guarantees that the assignment (ω, w) ↦ (v ↦ ω(v) w) is bilinear in ω and w separately, which is precisely what licenses the linear extension defining Θ over all of V* ⊗ W, including sums of simple tensors that are not themselves simple. Additivity preservation for general elements of Hom(V, W), not just the rank-one maps directly built from a single ω ⊗ w, rests on this bilinearity together with the fact that every element of V* ⊗ W can be written as a finite sum of simple tensors.

Consequences of Additivity Preservation

Zero Map Corresponds to Zero Tensor

The additive identity of Hom(V, W), the zero map sending every vector to 0 ∈ W, corresponds under Θ to the zero element of V* ⊗ W, since Θ is linear and every linear map sends the additive identity of its domain to the additive identity of its codomain. This matching of zero elements is a basic but necessary consequence of additivity preservation, confirming that the correspondence respects the full additive group structure, not merely nonzero elements.

Additive Inverses Correspond

Because Θ is linear, Θ(-α) = -Θ(α), so the tensor representing the negative of a linear map is the negative of the tensor representing the original map; additivity preservation extends automatically to preservation of additive inverses, completing the correspondence between the additive group structure of Hom(V, W) and that of V* ⊗ W.