# 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 notions above can be dualized by considering the notions of [alpha]-limit set of a point x, negatively Poisson stable points, et cetera.
1 Monitoring progress toward reaching goals of an indivi- dualized physical activity plan 54.
He was further apprised that Ataturk Avenue (West) Embassy Road would be dualized from Khayaban-e-Suhurwardhy to Khayaban-e-Margalla (Ayub Chowk).
He informed the meeting that the existing road was single carriage which will be dualized and improved.
The outputs will be (i) approximately 411 km of upgraded and dualized railway track for the Lahore-Peshawar section of ML-1 together with new signaling and telecommunications system (including power supply for these systems) and upgraded passenger facilities at Lahore, Rawalpindi, and Peshawar stations; (ii) 52 km of newly constructed double-track rail line linking Kaluwal and Pindora; and (iii) PR~s modernized and IT-based accounting system and PR's accounting data and information transformed and migrated into the new accounting system.
Considering the transposed and dualized spreads, we also obtain the following corollary
ISLAMABAD -- Capital Development Authority (CDA) has started construction work on dualized service road North and service road West of Sector E-12/2.