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 ?
We first define the acceptance condition in terms of a Rabin acceptance condition, as in Bhat and Cleaveland [1996a] (which dualizes the Streett condition in Emerson and Jutla [1991]).
The Ministry of Transport and Communications (MOTC) plans to dualize the fifth part of the Adam to Thumrait Road, a stretch of the Nizwa to Thumrait Road in Oman.
The Ministry of Transport and Communications (MOTC) plans to dualize the third part of the Adam to Thumrait Road, a stretch of the Nizwa to Thumrait Road in Oman.
ISLAMABAD -- National Highway Authority (NHA) is making concert measures to dualize Jamshoro-Sehwan Sharif National Highway road in order to minimise chances of accidents.
Ahsan Iqbal said the JCC also consented to dualize the Western route of CPEC : from Gwadar to Quetta which would be completed by the end of next year.
ISLAMABAD -- Minister for Parliamentary Affairs Sheikh Aftab said Tuesday that measures were being taken to dualize Jamshoro-Sehwan Sharif National Highway in order to minimise chances of accidents.
He informed the Committee that the capital is expanding with the passage of time and CDA has chalked out viable plans to develop the infrastructure, dualize the roads and construct under passes and flyovers besides improving the sewerage system and provision of recreational facilities.