# Dual

[doo-uh l, dyoo-] /ˈdu əl, ˈdyu-/

**adjective**

1.

of, relating 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.

**noun**, Grammar.

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.

/ˈdjuːəl/

**adjective**

1.

relating to or denoting two

2.

twofold; double

3.

(in the grammar of Old English, Ancient Greek, and certain other languages) denoting a form of a word indicating that exactly two referents are being referred to

4.

(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

**noun**

5.

(grammar)

**verb** duals, dualling, dualled

6.

(transitive) (Brit) to make (a road) into a dual carriageway

**adj.**

c.1600, from Latin dualis, from duo “two” (see two). Related: Dually.

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)

Tagged: d

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