3.15.1 Tensor Linear Map Dual Relation

Explore how tensor linear maps interact with their duals, revealing deep connections in algebraic structures and functional relationships.

Tensor Linear Map Dual Relation is the correspondence that associates to every linear map T: V → W between vector spaces a companion linear map T*: W* → V* between their dual spaces, running in the reverse direction, defined by precomposing covectors on W with T itself. This relation shows that dualization is not confined to individual vector spaces but extends to the linear maps connecting them, producing a systematic reversal of arrows that plays a central role in relating tensor constructions built on V to those built on W.

Definition of the Dual Map

The Precomposition Formula

Given a linear map T: V → W, its dual map T*: W* → V* is defined by sending a covector ψ in W* to the covector T*(ψ) in V* given by:

T^{*} (ψ) = ψ \circ T

meaning T*(ψ) is the functional on V obtained by first applying T to send a vector of V into W, and then applying ψ to extract a scalar. Explicitly, for any v in V:

(T^{*}, (ψ)) (v) = ψ (T, (v))

Why T(ψ) Lands in V

The composite ψ ∘ T is a function from V to the underlying field, and it is linear because it is a composition of two linear maps, T followed by ψ. A composition of linear maps is always linear, so T*(ψ) genuinely belongs to V*, confirming the dual map is well defined.

Direction Reversal

The dual map runs in the opposite direction from the original: T goes from V to W, while T* goes from W* to V*. This reversal is unavoidable given the formula, since T*(ψ) is built by having T act first, which requires starting in V, so the resulting functional naturally belongs to V* rather than W*.

Linearity of the Dual Map Construction

T* Is Linear in ψ

For fixed T, the map ψ ↦ T*(ψ) is itself linear: for covectors ψ_1, ψ_2 in W* and scalars a, b,

T^{*} (a ψ_{1} + b ψ_{2}) = a \cdot T^{*} (ψ_{1}) + b \cdot T^{*} (ψ_{2})

which follows directly from the linearity of composition with T, since (aψ_1 + bψ_2) ∘ T = a(ψ_1 ∘ T) + b(ψ_2 ∘ T) as a pointwise identity of functions on V.

Assignment of T to T* Is Also Linear

Beyond linearity in ψ, the overall assignment sending a linear map T to its dual T* is itself linear as a map between spaces of linear maps: for linear maps T_1, T_2: V → W and scalars a, b, the dual of aT_1 + bT_2 equals aT_1* + bT_2*. This double layer of linearity is part of what makes dualization a well-behaved algebraic operation rather than merely a formal trick applied one map at a time.

Composition and the Reversal of Order

Dual of a Composite Map

If T: V → W and S: W → U are linear maps, their composite S ∘ T: V → U has a dual map that reverses the order of composition:

(S \circ T) * = T^{*} \circ S^{*}

Verifying this directly: for ω in U* and v in V, (S∘T)*(ω)(v) = ω(S(T(v))), while (T*∘S*)(ω)(v) = T*(S*(ω))(v) = S*(ω)(T(v)) = ω(S(T(v))), matching exactly.

Contravariant Functoriality

This order-reversing behavior is described as contravariant functoriality: dualization sends the composite S ∘ T to T* ∘ S* rather than S* ∘ T*, the order one would expect from a direction-preserving construction. This contravariance is a defining structural feature of dualization, distinguishing it from constructions, such as the double dual embedding, that preserve the order of composition.

Dual of the Identity Map

The identity map id_V: V → V has dual (id_V)* = id_{V*}, the identity map on V*, since precomposing any covector with the identity leaves it unchanged. This, together with the order-reversing composition rule, confirms that dualization behaves consistently with the basic requirements expected of a systematic construction on linear maps.

Matrix Description of the Dual Map

The Transpose Relationship

Given bases of V and W, a linear map T: V → W is represented by a matrix M. Relative to the corresponding dual bases of V* and W*, the dual map T* is represented by the transpose matrix M^T. This is the source of the common alternative name "transpose map" for T*, since in coordinates the dual map is computed by transposing the matrix of the original map.

Consistency with Composition

The matrix identity (NM)^T = M^T N^T for matrices M representing T and N representing S mirrors exactly the composition rule (S∘T)* = T*∘S* derived abstractly, confirming that the coordinate description of the dual map is consistent with its basis-independent definition.

Diagrammatic Summary

The diagram places V and W on top with T pointing rightward between them, and V* and W* below with T* pointing leftward, visually capturing how dualization reverses the direction of the original map while operating between the corresponding dual spaces.