du⋅al
[doo-uh
l, dyoo-] 
| 1. | of, pertaining to, or noting two. |
| 2. | composed or consisting of two people, items, parts, etc., together; twofold; double: dual ownership; dual controls on a plane. |
| 3. | having a twofold, or double, character or nature. |
| 4. | Grammar. being or pertaining to a member of the category of number, as in Old English, Old Russian, or Arabic, that denotes two of the things in question. |
| 5. | the dual number. |
| 6. | a form in the dual, as Old English git “you two,” as contrasted with ge “you” referring to three or more. |
Based on the Random House Dictionary, © Random House, Inc. 2009.
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| du·al (dōō'əl, dyōō'-) adj.
[Latin duālis, from duo, two; see dwo- in Indo-European roots.] du'al·ly adv. |
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Dual
Du"al\, a. [L. dualis, fr. duo two. See Two.] Expressing, or consisting of, the number two; belonging to two; as, the dual number of nouns, etc., in Greek. Here you have one half of our dual truth. --Tyndall.Cite This Source
dual
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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.
(1997-03-16)
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