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
Collins Discovery Encyclopedia, 1st edition © HarperCollins Publishers 2005

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.
An Illustrated Dictionary of Aviation Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved

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.
This article is provided by FOLDOC - Free Online Dictionary of Computing (foldoc.org)
References in periodicals archive ?
Earlier, the governor inaugurated the dualised Bisola Durosinmi Etti/Hakeem Dickson Road, Lekki in Eti-Osa Local Government Area of Lagos, saying that his administration only prioritised renewal and development of road infrastructure but utilised an all-inclusive approach that had touched both rural and urban areas in the state.
Ilyas Khan an area resident said, one wonders that recently Lehtrar road was dualised but the approach roads are in dilapidated condition.
On a point, he was further informed that the road right upto Kohat and Fatehjang will also be dualised under a separate project later on.
He did not tell us why the Ninth Mile Corner - Markudi road has not been dualised like many other roads up North.'
Following the completion of this project, the link between Muscat and Salalah will be fully dualised along its 1,000 kilometre length.
Nineteen kilometre long road from Sakrand to Shaheed Benazirabad is also being dualised and progress on the project is about 60 per cent.84 Km Gharo-Keti Bander road is also being built in three packages and is expected to be completed by December 2011.
Obasanjo portends to love a real Nigeria, is it the real Nigeria that he ruled for eight years that could not make him as head of state or a permanent secretary of the Ministry of Roads and Transport think that the road to Ota should be tarred just like Buhari's town is now being dualised. If Daura can be dualised and finished in no time, why in eight years was Obasanjo not able to do the same for his own town?
The road being dualised in Oyo for almost eight years was yet to be completed and the quality of the road was bad compared to the quality of the roads in other zones of the state.
Responding to Usman Tarkai Parliament Secretary said that Peshawar Torkham Expressway would be dualised and construction work over the project would be started in 2011.
To start with, the present road networks leading from Sand Field, Maroko in Victoria Island, to Eleko junction, though, prides as 'Lekki-Expressway', in nothing but a glorified dualised face-me-i-face-you road, common in the neighborhood within the metropolis.