# dual homomorphism

## Definition.

Let $U,V$ be vector spaces^{} over a field
$\mathbb{K}$, and $T:U\to V$ be a homomorphism^{} (a
linear map) between them.
Letting ${U}^{*},{V}^{*}$ denote the corresponding dual
spaces^{}, 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 $T$ relative
to 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 the
dualizing operation^{} as the composition^{} of the following canonical
isomorphisms:

$$Hom(U,V)\stackrel{\simeq}{\u27f6}{U}^{*}\otimes V\stackrel{\simeq}{\u27f6}{({V}^{*})}^{*}\otimes {U}^{*}\stackrel{\simeq}{\u27f6}Hom({V}^{*},{U}^{*}).$$ |

## Category theory perspective.

The dualizing operation behaves contravariantly 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 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 $M\in {Mat}_{n,m}(\mathbb{K})$ 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 ${M}^{t}\in {Mat}_{m,n}(\mathbb{K})$ relative to the corresponding dual bases of ${U}^{*},{V}^{*}$.

Title | dual homomorphism |

Canonical name | DualHomomorphism |

Date of creation | 2013-03-22 12:29:33 |

Last modified on | 2013-03-22 12:29:33 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 10 |

Author | rmilson (146) |

Entry type | Definition |

Classification | msc 15A72 |

Classification | msc 15A04 |

Synonym | adjoint homomorphism |

Synonym | adjoint |

Related topic | LinearTransformation |

Related topic | DualSpace |

Related topic | DoubleDualEmbedding |