Page 163 - A Course in Linear Algebra with Applications
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5.3:  Operations  with  Subspaces           147


             Our  last  example  of  coset  formation  is  a  geometric  one.
        Example     5.3.10

        Let  A  and  B  be  vectors  in  R  3  representing  non-parallel  line
        segments  in  3-dimensional  space.  Then  the  subspace

                                U=<A,        B>


        has  dimension  2 and  consists  of  all  cA  + dB,  (c, d  E  R).  The
        vectors  in  U are represented  by  line segments,  drawn  from  the
        origin,  which  lie  in  a  plane  P.  Now  choose  X  G  R  ,  with
                           T
        X  =  (xi,  x 2,  x 3) .
             A  typical  vector  in the  coset  X  + U  has  the  form
        X  + cA  +  dB,  with  c, d  e  R,  i.e.,


                                                                 T
              (xi  + cai  + dbi  x 2  + ca 2  + db2  X3  + ca 3  +  db 3) .

                         (
        Now the   points xi+cai+d&i,     X2+ca2+db 2,    Xs + cas + db 3)
        lie  in  the  plane  Pi  passing  through  the  point  (xi,  x 2,  £3),
        which  is  parallel  to  the  plane  P.  This  is  seen  by  forming  the
        line  segment  joining  two  such  points.  The  elements  of  X  + U
        correspond  to  the  points  in the  plane  Pi:  the  latter  is  called
        a  translate  of the  plane  P.
        Dimension     of  a  Quotient  Space

             We  conclude  the  discussion  by  noting  a  simple  formula
        for  the  dimension  of  a  quotient  space  of  a  finite  dimensional
        vector  space.

        Theorem     5.3.7
        Let  U  be  a  subspace  of  a  finite dimensional  vector  space  V.
         Then
                      dim(V/C/)   =   dim(y)   -  dim(£/).
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