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dual homomorphism
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(Definition)
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Let $U,V$ be vector spaces over a field $\kfield$ , and $T:U\rightarrow V$ be a homomorphism (a linear map) between them. Letting $U\dual, V\dual$ denote the corresponding dual spaces, we define the dual homomorphism $T\dual:V\dual\rightarrow U\dual$ , to be the linear mapping with action
$$\alpha \to \alpha\circ T,\quad \alpha\in V^*.$$
We can also characterize $T^*$ as the adjoint of $T$ relative to the natural evaluation bracket between linear forms and vectors: $$\left<-,-\right>_U: U\dual\times U\rightarrow \kfield,\qquad \left<\alpha,u\right> = \alpha(u),\quad \alpha\in U\dual,\; u\in U.$$ To be more precise $T\dual$ is characterized by the condition $$\left<T^*\alpha,u\right>_U = \left< \alpha,Tu\right>_V ,\quad \alpha\in V\dual,\; u\in U.$$
If $U$ and $V$ are finite dimensional, we can also characterize the dualizing operation as the composition of the following canonical isomorphisms: $$\Hom(U,V) \stackrel{\simeq}{\longrightarrow} U\dual\otimes V \stackrel{\simeq}{\longrightarrow} (V^*)^*\otimes U^* \stackrel{\simeq}{\longrightarrow} \Hom(V^*,U^*). $$
The dualizing operation behaves contravariantly with respect to composition, i.e. $$(S\circ T)^* = T\dual \circ S\dual,$$ 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.
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 $M\in \Mat_{n,m}(\kfield)$ be the matrix of $T$ relative to some fixed bases of $U$ and $V$ . Then, the dual homomorphism $T\dual$ is represented as the transposed matrix $M\supt\in\Mat_{m,n}(\kfield)$ relative to the corresponding dual bases of $U\dual, V\dual$ .
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"dual homomorphism" is owned by rmilson. [ full author list (2) ]
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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
There are 11 references to this entry.
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 8518 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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Pending Errata and Addenda
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