3.15.2 Tensor Dual Map Domain Reversal

Tensor Dual Map Domain Reversal flips the domain of a linear map, linking dual spaces through contravariant transformation in tensor algebra.

Tensor Dual Map Domain Reversal is the property by which the dual of a linear map exchanges the roles of source and target, so that a map f: V → W induces a dual map f*: W* → V* whose domain is the dual of f's codomain and whose codomain is the dual of f's domain. This reversal is the defining structural signature of duality on linear maps: it is not an incidental feature but a necessary consequence of how covectors act on vectors through precomposition, and it governs every downstream construction built from dual maps, including their composition rule, their matrix representation, and their behavior on tensor products.

Why the Domain Reverses

Precomposition Forces the Reversal

A covector φ in W* is a linear functional defined on W, so it can only be evaluated on elements of W. To turn φ into a functional on V, the only available operation is to first map a vector of V into W using f, and then apply φ. This composite φ ∘ f is a functional on V, which defines f*(φ):

f^{*} (φ) = φ \circ f

Since φ was taken from W*, the input of f* lives in W*, and since the output φ ∘ f is a functional on V, the output of f* lives in V*. The map f* therefore runs from W* to V*, the reverse of the direction of f.

No Natural Forward Map Exists

Without additional structure such as an inner product or an explicit inverse, there is no natural way to push a covector on V forward to a covector on W using f alone, because f may fail to be injective or surjective. Domain reversal is therefore not a matter of choice: it is the only direction in which a canonical map between dual spaces can be defined purely from f.

Formal Statement of the Reversal

Domain and Codomain Exchange

If f has domain V and codomain W, then f* has domain W* and codomain V*:

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

This exchange is total: every occurrence of V in the data of f corresponds to V* appearing as the codomain of f*, and every occurrence of W corresponds to W* appearing as the domain of f*.

Effect on Composition Order

Because the domain and codomain are exchanged, composing two dual maps requires reversing the order in which the original maps were composed. For f: V → W and g: W → U, the composite g ∘ f: V → U has dual

(g \circ f)^{*} = f^{*} \circ g^{*}

which is only well-defined because f* starts where g* ends: g* maps U* → W* and f* maps W* → V*, so the composite f* ∘ g* runs from U* to V*, matching the reversed domain and codomain of (g ∘ f)*.

Consequences for Matrix Representations

Transposition as the Coordinate Shadow of Reversal

When f is represented by an m × n matrix A relative to bases of V (dimension n) and W (dimension m), the domain reversal of f* is reflected in the shape of its matrix: f* is represented by the n × m transpose A^T, which maps coordinate vectors of length m (representing elements of W*) to coordinate vectors of length n (representing elements of V*). The swapped matrix dimensions are the numerical expression of the swapped domain and codomain.

Rank Preservation Across the Reversal

Although the domain and codomain are exchanged, the rank of the map is preserved: f and f* have the same rank, equal to the rank of A and A^T. Domain reversal changes where the map starts and ends, but it does not change the dimension of its image, since row rank and column rank of a matrix always coincide.

Domain Reversal in Tensor Contexts

Reversal on Contravariant Slots

When the dual map structure acts on a type (p, q) tensor built from V and V*, the domain reversal governs which factors are affected by f and which are affected by f*: the p contravariant slots, drawn from V, are acted on directly by f, while the q covariant slots, drawn from V*, are acted on by f*, whose reversed domain of V*-valued arguments matches exactly the covariant slots of the tensor.

Reversal and the Double Dual

Applying domain reversal twice returns to the original direction: the dual of f*: W* → V* is a map (f*)*: V** → W**, and under the canonical identification of a finite-dimensional space with its double dual, this composite recovers f: V → W. Domain reversal is therefore an involution at the level of direction, even though it is applied through an intermediate space at each step.

Diagrammatic Summary

The diagram shows that what serves as the domain for f corresponds to the codomain for f*, and what serves as the codomain for f corresponds to the domain for f*, making explicit the exchange of roles that defines domain reversal.