2.22.2 Tensor Linear Map Codomain Space

The codomain of a tensor linear map is the target space where tensor transformations act, key to understanding tensor algebra structure and mapping behavior.

Tensor Linear Map Codomain Space is the vector space W that supplies the output side of a linear map φ : V → W, considered specifically in its role inside the tensor identification Hom(V, W) ≅ V* ⊗ W, where the codomain space contributes itself directly, undualized, to the tensor product representing the map. The codomain's role is the mirror image of the domain's: where the domain space enters through its dual to consume an input vector, the codomain enters unchanged to produce an output vector, and this asymmetry is what fixes the variance type of each index in the tensor description of a linear map.

Why the Codomain Contributes Itself, Undualized

Producing a Vector Requires No Dualizing

A rank-one linear map ω ⊗ w, built from ω ∈ V* and w ∈ W, acts on an input v by (ω ⊗ w)(v) = ω(v) w, and the output of this action is a scalar multiple of w itself, an element of W, not of W* or any dualized version of W. Because the output of a linear map is simply a vector in the codomain, the codomain space is exactly what is needed, with no dualizing required, to supply that output.

(ω \otimes w) (v) = ω (v) \cdot w \in W

Contravariant Role of the Codomain Index

Because the codomain contributes directly rather than through a dual, its associated index in the (1, 1)-tensor representation of φ is an upper, contravariant index, matching the general rule that upper indices represent vector-valued, output-producing slots of a tensor, while lower indices represent covector-valued, input-consuming slots.

Dimension and Basis of the Codomain Space

Dimension Contribution

If dim(W) = m, the codomain contributes a factor of m to the total dimension dim(V* ⊗ W) = dim(V) × dim(W) of the space of linear maps from V to W; a linear map from an n-dimensional domain to an m-dimensional codomain has exactly nm independent scalar components, matching the familiar count of entries in an m × n matrix.

Codomain Basis in the Matrix Description

Fixing a basis {f_j} for the codomain W, alongside the dual basis {e^i} induced by a basis {e_i} of the domain, produces the standard basis {e^i ⊗ f_j} of V* ⊗ W. The upper index j of a matrix entry a^j_i is precisely the index labeling this codomain basis, so each row of a matrix, in the usual convention where the upper index selects the output coordinate, corresponds to one basis vector of the codomain.

Effect of a Change of Codomain Basis

Transformation of the Contravariant Slot

If the codomain basis is changed from {f_j} to {f'_j} via a transition matrix C, the codomain-associated upper index of the tensor a^j_i transforms with the inverse C^{-1}, in accordance with the standard contravariant transfer rule applied to output coordinates:

a^{'} j i = \sum_{k} {(C^{- 1})}_{k}^{j} a_{i}^{k}

Independence From the Domain Basis

Since V* and W are separate tensor factors, a change of codomain basis affects only the upper index of the matrix of φ, leaving the lower, domain-associated index unaffected. This independence lets the domain and codomain bases be changed separately, with the matrix of φ updating through the corresponding factor of V* ⊗ W only.

The Codomain Space in Higher Multilinear Constructions

Codomain of a Multilinear Map

For a k-linear map V × ⋯ × V → W, identified with an element of (V*)^{⊗k} ⊗ W, the codomain contributes exactly one undualized factor regardless of how many domain arguments the map takes, since the map still produces a single output vector in W no matter how many inputs it consumes. This asymmetry, many dual domain factors against one direct codomain factor, is visible directly in the index structure: a k-linear map into W is described by a tensor with k lower indices and one upper index.

Composability and the Codomain-Domain Match

When two linear maps φ : U → V and ψ : V → W are composed, the codomain V of φ must equal the domain V of ψ in order for the tensor contraction defining ψ ∘ φ to make sense, since contraction pairs the upper, codomain-derived index of φ's tensor α ∈ U* ⊗ V against the lower, domain-derived index of ψ's tensor β ∈ V* ⊗ W. The codomain space is therefore not only the target of a single map but also the shared space that makes chains of composable maps well defined.