3.15 Tensor Dual Map Structure

Tensor Dual Map Structure is the algebraic framework describing how linear maps between vector spaces induce corresponding linear maps between their dual spaces, and how this correspondence extends coherently to tensors built from those spaces. Given a linear map f: V → W between finite-dimensional vector spaces, the dual map structure produces a map f*: W* → V* running in the opposite direction, called the pullback or transpose map, together with rules governing how f and f* act on tensor products, contractions, and mixed tensors of type (p, q). This structure is what allows operations on vectors to be systematically translated into operations on covectors and higher tensors while preserving all algebraic relations.

The Dual Map Construction

Definition of the Pullback

Given a linear map f: V → W, the dual map f*: W* → V* is defined by precomposition: for any covector φ ∈ W*, the image f*(φ) is the covector in V* obtained by first applying f and then φ.

for every v ∈ V. This equation characterizes f* completely: it is the unique linear map making the composite φ ∘ f equal to f*(φ) applied directly.

Reversal of Direction

The defining feature of the dual map is contravariance: while f sends V to W, its dual f* sends W* to V*, the reverse direction. This reversal is not a matter of convention but a structural consequence of precomposition, since covectors on the target W can only be pulled back to covectors on the source V through f, never pushed forward without additional data such as an inverse or an inner product.

Matrix Representation

If f is represented by a matrix A relative to chosen bases of V and W, then f* is represented by the transpose matrix A^T relative to the corresponding dual bases. This is the origin of the term "transpose map": dualizing a linear map at the abstract level corresponds exactly to transposing its matrix at the level of coordinates.

Functorial Properties

Contravariant Functoriality

The assignment V ↦ V* and f ↦ f* defines a contravariant functor from the category of vector spaces to itself. Contravariance means the functor reverses the order of composition:

for composable linear maps f: V → W and g: W → U. Applying the dual construction to a composite g ∘ f yields f* ∘ g*, with the order of the factors reversed.

Identity Preservation

The dual of the identity map id_V: V → V is the identity map id_{V*}: V* → V*. Together with the reversal-of-composition rule, this confirms that duality is a genuine contravariant functor rather than merely a pointwise construction on individual maps.

Behavior Under Isomorphism

If f: V → W is an isomorphism, then f*: W* → V* is also an isomorphism, and (f*)^{-1} = (f^{-1})*. Dualizing therefore preserves invertibility while continuing to reverse direction, so an isomorphism of vector spaces induces an isomorphism of their dual spaces.

Extension to Tensor Products

Dual Maps on Simple Tensors

The dual map structure extends to tensor products by acting on each factor. Given linear maps f: V → W and g: U → X, there is an induced dual map on the tensor product of the dual spaces, acting on simple tensors by

and extended by linearity to all of the tensor product space. This shows that dualization is compatible with the tensor product construction, factor by factor.

Action on Mixed Type (p, q) Tensors

For an endomorphism f: V → V, the induced action on the space T^p_q(V) of type (p, q) tensors applies f to each of the p contravariant slots and f* to each of the q covariant slots simultaneously. This mixed action is what allows a single linear transformation of V to consistently transform every tensor built from V and V*, regardless of its valence.

Compatibility with Contraction

The dual map structure is compatible with tensor contraction: applying f and f* to a tensor and then contracting a contravariant slot against a covariant slot gives the same result as contracting first and then transforming, provided the contraction pairs a slot acted on by f with a slot acted on by f*. This compatibility is what makes contraction, such as the trace of a (1,1) tensor, a well-defined operation independent of how the ambient linear maps are chosen.

The Natural Pairing and Double Duality

The Canonical Bilinear Pairing

The dual map structure rests on the canonical pairing between V and V*, the bilinear evaluation map

given by ⟨φ, v⟩ = φ(v). The dual map f* is precisely the adjoint of f with respect to this pairing, satisfying ⟨f*(φ), v⟩ = ⟨φ, f(v)⟩ for all φ ∈ W* and v ∈ V.

Double Dual and Evaluation

For a finite-dimensional space V, there is a canonical isomorphism V ≅ V** sending each vector v to the evaluation functional ev_v: V* → F, ev_v(φ) = φ(v). Under this identification, dualizing a dual map twice recovers the original map: (f*)* = f, confirming that the dual map structure is an involution at the level of the double dual.

Consistency of the Whole Structure

Taken together, the pullback construction, its contravariant functoriality, its extension to tensor products and contractions, and its involutive behavior under double duality form a single coherent structure. This structure is what guarantees that any algebraic identity involving vectors, covectors, and tensors built from a linear map f remains valid after transporting every object through f and f* in the appropriate directions.

Diagrammatic Summary

The diagram shows the linear map f running forward from V to W, and the induced dual map f* running backward from W* to V*, illustrating the contravariant reversal that defines the tensor dual map structure.