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dual homomorphism (Definition)

Definition.

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

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

$\displaystyle \left<-,-\right>_U: U^*\times U\rightarrow \mathbb{K},\qquad \left<\alpha,u\right> = \alpha(u),\quad \alpha\in U^*,\; u\in U.$
To be more precise $ T^*$ is characterized by the condition
$\displaystyle \left<T^*\alpha,u\right>_U = \left< \alpha,Tu\right>_V ,\quad \alpha\in V^*,\; u\in U.$

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

$\displaystyle \mathop{\mathrm{Hom}}\nolimits (U,V) \stackrel{\simeq}{\longright... ...* \stackrel{\simeq}{\longrightarrow} \mathop{\mathrm{Hom}}\nolimits (V^*,U^*). $

Category theory perspective.

The dualizing operation behaves contravariantly with respect to composition, i.e.
$\displaystyle (S\circ T)^* = T^*\circ S^*,$
for all vector space homomorphisms $ S, T$ with suitably matched domains. Furthermore, the dual of the identity homomorphism is the identity homomorphism of the dual space. Thus, using the language of category theory, the dualizing operation can be characterized as the homomorphism action of the contravariant, dual-space functor.

Relation to the matrix transpose.

The above properties closely mirror 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 calculated by taking the transpose.

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



"dual homomorphism" is owned by rmilson. [ full author list (2) ]
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See Also: linear transformation, dual space, double dual embedding

Other names:  adjoint homomorphism, adjoint

Attachments:
dual homomorphism of the derivative (Example) by rmilson
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Cross-references: bases, fixed, finite-dimensional, level, transpose, matrix, algebraic, properties, functor, category theory, language, identity, domains, isomorphisms, canonical, composition, operation, finite dimensional, vectors, linear forms, action, dual spaces, linear map, homomorphism, field, vector spaces
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This is version 7 of dual homomorphism, born on 2002-02-26, modified 2006-08-03.
Object id is 2717, canonical name is DualHomomorphism.
Accessed 7266 times total.

Classification:
AMS MSC: 15A04 (Linear and multilinear algebra; matrix theory :: Linear transformations, semilinear transformations)
 15A72 (Linear and multilinear algebra; matrix theory :: Vector and tensor algebra, theory of invariants)
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Relationship of various notions of adjoint by archibal on 2004-04-02 00:52:38
We list many different notions of adjoint; four in linear algebra, for example. There's also a categorical notion of adjoint. I would like to see some explanation of how all these notions are related. So I guess I have two questions:

* How *are* they related? I can see several relationships between some of them, but for example: is the adjoint homomorphism a categorical adjoint? Does it share some properties? Is it just a name in linear algebra for something which needs no name in category theory, and there's no relation between the concepts?

* Where should the descriptions go? Probably some should go everywhere; the most technical should indicate what they're generalizing, the most elementary should indicate what they are special cases of, the ones that are named by analogy should explain the analogy and its limits, and the genuinely unrelated ones should say so.
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