dual homomorphism

Definition.

Let $U,V$ be vector spaces over a field$𝕂$, and $T:U\to V$ be a homomorphism (alinear map) between them.Letting $U^{*},V^{*}$ denote the corresponding dualspaces, we define the dual homomorphism$T^{*}:V^{*}\to U^{*}$,to be the linear mapping with action

$$\alpha \to \alpha \circ T,\alpha \in V^{*}.$$

We can also characterize $T^{*}$ as the adjoint of $T$ relativeto the natural evaluation bracket between linear forms and vectors:

To be more precise $T^{*}$ is characterized by the condition

If $U$ and $V$ are finite dimensional, we can also characterize thedualizing operation as the composition of the following canonicalisomorphisms:

$$Hom(U,V)\stackrel{\simeq }{⟶}{U}^{*}\otimes V\stackrel{\simeq }{⟶}{(V^{*})}^{*}\otimes U^{*}\stackrel{\simeq }{⟶}Hom(V^{*},U^{*}).$$

Category theory perspective.

The dualizing operation behavescontravariantly with respect to composition, i.e.

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

for all vector space homomorphisms $S,T$ with suitably matched domains. Furthermore, the dual of theidentity homomorphism is the identity homomorphism of the dual space.Thus, using the language of category theory, the dualizing operationcan be characterized as the homomorphism action of the contravariant,dual-space functor.

Relation to the matrix transpose.

The above properties closelymirror the algebraic properties of the matrix transpose operation.Indeed, $T^{*}$ is sometimes referred to as the transpose of $T$,because at the level of matrices the dual homomorphism is calculatedby taking the transpose.

To be more precise, suppose that $U$ and $V$ are finite-dimensional,and let $M\in {Mat}_{n,m}(𝕂)$ be the matrix of $T$ relative tosome fixed bases of $U$ and $V$. Then, the dual homomorphism $T^{*}$is represented as the transposed matrix $M^t\in {Mat}_{m,n}(𝕂)$relative to the corresponding dual bases of $U^{*},V^{*}$.