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algebra

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algebra


  4  definitions  found 
 
  From  Webster's  Revised  Unabridged  Dictionary  (1913)  [web1913]: 
 
  Algebra  \Al"ge*bra\,  n.  [LL.  algebra,  fr  Ar  al-jebr  reduction 
  of  parts  to  a  whole,  or  fractions  to  whole  numbers,  fr 
  jabara  to  bind  together,  consolidate;  al-jebr 
  w'almuq[=a]balah  reduction  and  comparison  (by  equations):  cf 
  F.  alg[`e]bre,  It  &  Sp  algebra.] 
  1.  (Math.)  That  branch  of  mathematics  which  treats  of  the 
  relations  and  properties  of  quantity  by  means  of  letters 
  and  other  symbols.  It  is  applicable  to  those  relations 
  that  are  true  of  every  kind  of  magnitude. 
 
  2.  A  treatise  on  this  science. 
 
  From  Webster's  Revised  Unabridged  Dictionary  (1913)  [web1913]: 
 
  Mathematics  \Math`e*mat"ics\,  n.  [F.  math['e]matiques,  pl.,  L. 
  mathematica,  sing.,  Gr  ?  (sc.  ?)  science.  See  {Mathematic}, 
  and  {-ics}.] 
  That  science,  or  class  of  sciences,  which  treats  of  the  exact 
  relations  existing  between  quantities  or  magnitudes,  and  of 
  the  methods  by  which  in  accordance  with  these  relations, 
  quantities  sought  are  deducible  from  other  quantities  known 
  or  supposed;  the  science  of  spatial  and  quantitative 
  relations. 
 
  Note:  Mathematics  embraces  three  departments,  namely:  1. 
  {Arithmetic}.  2.  {Geometry},  including  {Trigonometry} 
  and  {Conic  Sections}.  3.  {Analysis},  in  which  letters 
  are  used  including  {Algebra},  {Analytical  Geometry}, 
  and  {Calculus}.  Each  of  these  divisions  is  divided  into 
  pure  or  abstract,  which  considers  magnitude  or  quantity 
  abstractly,  without  relation  to  matter;  and  mixed  or 
  applied,  which  treats  of  magnitude  as  subsisting  in 
  material  bodies,  and  is  consequently  interwoven  with 
  physical  considerations. 
 
  From  WordNet  r  1.6  [wn]: 
 
  algebra 
  n  :  the  mathematics  of  generalized  arithmetical  operations 
 
  From  The  Free  On-line  Dictionary  of  Computing  (13  Mar  01)  [foldoc]: 
 
  algebra 
 
    1.  A  loose  term  for  an  {algebraic 
  structure}. 
 
  2.  A  {vector  space}  that  is  also  a  {ring},  where  the  vector 
  space  and  the  ring  share  the  same  addition  operation  and  are 
  related  in  certain  other  ways. 
 
  An  example  algebra  is  the  set  of  2x2  {matrices}  with  {real 
  numbers}  as  entries,  with  the  usual  operations  of  addition  and 
  matrix  multiplication,  and  the  usual  {scalar}  multiplication. 
  Another  example  is  the  set  of  all  {polynomials}  with  real 
  coefficients,  with  the  usual  operations. 
 
  In  more  detail,  we  have: 
 
  (1)  an  underlying  {set}, 
 
  (2)  a  {field}  of  {scalars}, 
 
  (3)  an  operation  of  scalar  multiplication,  whose  input  is  a 
  scalar  and  a  member  of  the  underlying  set  and  whose  output  is 
  a  member  of  the  underlying  set  just  as  in  a  {vector  space}, 
 
  (4)  an  operation  of  addition  of  members  of  the  underlying  set 
  whose  input  is  an  {ordered  pair}  of  such  members  and  whose 
  output  is  one  such  member,  just  as  in  a  vector  space  or  a 
  ring, 
 
  (5)  an  operation  of  multiplication  of  members  of  the 
  underlying  set  whose  input  is  an  ordered  pair  of  such  members 
  and  whose  output  is  one  such  member,  just  as  in  a  ring. 
 
  This  whole  thing  constitutes  an  `algebra'  iff: 
 
  (1)  it  is  a  vector  space  if  you  discard  item  (5)  and 
 
  (2)  it  is  a  ring  if  you  discard  (2)  and  (3)  and 
 
  (3)  for  any  scalar  r  and  any  two  members  A,  B  of  the 
  underlying  set  we  have  r(AB)  =  (rA)B  =  A(rB).  In  other  words 
  it  doesn't  matter  whether  you  multiply  members  of  the  algebra 
  first  and  then  multiply  by  the  scalar,  or  multiply  one  of  them 
  by  the  scalar  first  and  then  multiply  the  two  members  of  the 
  algebra.  Note  that  the  A  comes  before  the  B  because  the 
  multiplication  is  in  some  cases  not  commutative,  e.g.  the 
  matrix  example. 
 
  Another  example  (an  example  of  a  {Banach  algebra})  is  the  set 
  of  all  {bounded}  {linear  operators}  on  a  {Hilbert  space},  with 
  the  usual  {norm}.  The  multiplication  is  the  operation  of 
  {composition}  of  operators,  and  the  addition  and  scalar 
  multiplication  are  just  what  you  would  expect. 
 
  Two  other  examples  are  {tensor  algebras}  and  {Clifford 
  algebras}. 
 
  [I.  N.  Herstein  "Topics_in_Algebra"]. 
 
  (1999-07-14) 
 
 




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