dual

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dual

Maths Logic (of structures or expressions) having the property that the interchange of certain pairs of terms, and usually the distribution of negation, yields equivalent structures or expressions

dual

i. Duplicated controls permitting either an instructor and a pupil or the pilot and the copilot to operate controls.
ii. Instructional flying hours or sorties logged in a flying log by the pupil.

dual

(mathematics)
Every field of mathematics has a different meaning of dual. Loosely, where there is some binary symmetry of a theory, the image of what you look at normally under this symmetry is referred to as the dual of your normal things.

In linear algebra for example, for any vector space V, over a field, F, the vector space of linear maps from V to F is known as the dual of V. It can be shown that if V is finite-dimensional, V and its dual are isomorphic (though no isomorphism between them is any more natural than any other).

There is a natural embedding of any vector space in the dual of its dual:

V -> V'': v -> (V': w -> wv : F)

(x' is normally written as x with a horizontal bar above it). I.e. v'' is the linear map, from V' to F, which maps any w to the scalar obtained by applying w to v. In short, this double-dual mapping simply exchanges the roles of function and argument.

It is conventional, when talking about vectors in V, to refer to the members of V' as covectors.
References in periodicals archive ?
The Ministry of Transport and Communications plans to build a 200 Km long, dualized road from Bidbid to Sur, in Oman.
Construction of a 143 km long dualized road from Masroun to Sur.
He said that 68 km Hyderabad-Mirpurkhas Section has already been dualized by Government of Sindh through Public-Private-Partnership (PPP) basis, but at present there is no proposal under consideration for dualization of Mirpurkhas -Umerkot Section.
Replying to a question, he said that western alignment would be dualized if traffic volume grew in future.
Ashfaq Ahmed, a resident of Tufail Road, said while the Army had widened and dualized numerous roads, it was strange that the Cantt officials were unable to carry out simple repair work and a number of roads in the area were turning into ponds due to lethargy of the civic body staff.
Hasnain Gardezi, an area resident, told APP that Lethrar Road has been dualized but all approach roads along Khanna Pull are in dilapidated condition for the last many years due to non-maintenance by the Authority.
A reorientation of agricultural support programmes, he said, would start with the recognition that South Africa is confronted with "a strongly dualized agricultural sector,"and that the current organization of markets has been inherited from the apartheid era.