3.15.3 Tensor Dual Map Codomain Reversal

Tensor Dual Map Codomain Reversal explains how dual maps reverse codomain in tensor algebra, connecting dual spaces through linear transformations.

Tensor Dual Map Codomain Reversal is the complementary aspect of dual map reversal that focuses on the fate of the codomain: given a linear map f: V → W, the space W, which serves as the target of f, does not remain the target after dualization. Instead, its dual W* becomes the source of the induced map f*, so the codomain of the original map is converted into the domain of the dual map. This shift complements domain reversal and together the two describe the full exchange of roles that occurs when a linear map is dualized.

The Codomain's New Role

From Target to Source

In the original map f: V → W, the space W receives the outputs of f; it is the target. Once dualized, W* no longer plays the role of a target for anything related to f*; instead, W* supplies the inputs to f*. An element φ ∈ W* is fed into f*, producing an element of V*:

f^{*} : W^{*} \to V^{*}

The space that once absorbed the outputs of f now, in dual form, generates the inputs of f*. This inversion of role, rather than a mere relabeling, is what the term "codomain reversal" captures.

Why the Codomain Cannot Stay a Codomain

If W* were to remain a codomain after dualization, there would need to be a canonical map landing in W*, built from f alone. No such map exists in general: f produces elements of W, not functionals on W, so there is no way to canonically construct an element of W* as an "output" associated with f. The only canonical operation available is precomposition, which uses elements of W* as inputs, converting them into elements of V*. This is why W* is forced into the role of a domain rather than a codomain.

Formal Description

Codomain of f Becomes Domain of f*

Writing the correspondence explicitly,

codomain (f) = W \Rightarrow domain (f^{*}) = W^{*}

This identity holds regardless of whether f is injective, surjective, or bijective: codomain reversal is a statement about the dual space construction itself, not about any property of f beyond its being a linear map with codomain W.

Interaction with Surjectivity and Injectivity

If f is surjective onto W, then f* is injective from W* into V*; if f is injective, then f* is surjective onto its image inside V*. These exchanges follow from codomain reversal together with domain reversal: properties tied to how f fills its codomain W translate into properties of how f* behaves on its domain W*.

Codomain Reversal in Matrix Form

Column Space Becomes Row Space Data

If f is represented by matrix A, the codomain W corresponds to the space in which the columns of A live, of dimension equal to the number of rows of A. After transposition, A^T has this same dimension as its number of columns, matching the fact that W*, now a domain, must have coordinate vectors of that length feeding into f*. The codomain's dimension becomes, after reversal, the length of the input vectors accepted by the dual map.

Consistency with Domain Reversal

Codomain reversal never occurs in isolation; it is always paired with domain reversal, since a linear map's dual is defined as a single map f*: W* → V*. The two reversals are two aspects of one construction: the codomain of f becoming the domain of f*, and the domain of f becoming the codomain of f*, together producing a map that runs in the fully reversed direction.

Codomain Reversal in Tensor Constructions

Effect on Covariant Slots

In the transformation of a type (p, q) tensor under an endomorphism f: V → V, the covariant slots of the tensor, valued in V*, are exactly the slots affected by f*. Since f* takes its domain from the dualized codomain of f, the covariant slots receive their transformation from the same reversed-direction map, ensuring that covariant and contravariant slots transform consistently within a single tensor.

Preserving Contraction Identities

Because the codomain of f becomes the domain of f* in a precise, canonical way, contractions between a contravariant slot transformed by f and a covariant slot transformed by f* remain well-defined after transformation. Codomain reversal guarantees that the covariant side of a contraction pairs correctly with the space that f* actually accepts as input.

Diagrammatic Summary

The diagram labels W explicitly as the codomain of f and W* explicitly as the domain of f*, making visible that the space which once received outputs from f becomes the space that supplies inputs to f*.