dual
    adj 1: consisting of or involving two parts or components
           usually in pairs; "an egg with a double yolk"; "a double
           (binary) star"; "double doors"; "dual controls for pilot
           and copilot"; "duple (or double) time consists of two (or
           a multiple of two) beats to a measure" [syn: double,
           dual, duple]
    2: having more than one decidedly dissimilar aspects or
       qualities; "a double (or dual) role for an actor"; "the
       office of a clergyman is twofold; public preaching and
       private influence"- R.W.Emerson; "every episode has its
       double and treble meaning"-Frederick Harrison [syn: double,
       dual, twofold, two-fold, treble, threefold, three-
       fold]
    3: a grammatical number category referring to two items or units
       as opposed to one item (singular) or more than two items
       (plural); "ancient Greek had the dual form but it has merged
       with the plural form in modern Greek"

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.
   [1913 Webster]

         Here you have one half of our dual truth. --Tyndall.
   [1913 Webster]

dual

    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)