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dual

## dual

```  3  definitions  found

From  Webster's  Revised  Unabridged  Dictionary  (1913)  [web1913]:

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.

From  WordNet  r  1.6  [wn]:

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},  {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},
{twofold},  {treble},  {threefold}]
3:  a  grammatical  number  category  referring  to  two  items  or
units  (as  opposed  to  one  item  (singular)  or  more  than  one
item  (plural));  "ancient  Greek  had  the  dual  form  but  it
has  merged  with  the  plural  form  in  modern  Greek"

From  The  Free  On-line  Dictionary  of  Computing  (13  Mar  01)  [foldoc]:

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)

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