Page 191 - A Course in Linear Algebra with Applications
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6.2:  Linear  Transformations  and  Matrices    175

       Hence  T  is  represented  with  respect  to  B'  by


                                       / 0  2   0 \
                                 - 1
                           [/At/   = 0      0   4   .
                                       \ 0  0   0 /


       This  conclusion  is  easily  checked.  An  arbitrary  element  of
                                                                   2
       P 3 (R)  can be written  in the  form  /  =  a(l)+6(2ir) + c(4a; -2).
       Then   it  is  claimed  that  the  coordinate  vector  of  T(f)  with
       respect  to the  basis  B'  is

                        / 0  2   0 \  / a \      / 2 6 \
                         0   0   4  6       ==     4 c .
                        \0   0   0/  \ c /         W

       This  is  correct  since

                                2
         26(1)  +  4c(2x)  +  0(4a?  -  2) =  2b +  8cx
                                                             2
                                      =  (a{l)  +  b(2x)  + c(4x  -  2))'.

        Similar  matrices
            Let  A  and  B  be  two  n  x  n  matrices  over  a  field  F;  then
        B  is  said  to  be  similar  to  A  over  F  if  there  is  an  invertible
       n  x  n  matrix  S  with  entries  in  F  such  that

                                             1
                                 B  =  SAS~ .

        Thus  the  essential  content  of  6.2.6  is that  two  matrices  which
        represent the same linear  operator  on  a finite-dimensional  vec-
        tor  space  are  similar.  Because  of this  fact  it  is to  be  expected
        that  similar  matrices  will  have  many  properties  in  common:
        for  example,  similar  matrices  have  the  same  determinant.  In-
        deed  if  B  =  SAS~\  then  by  3.3.3  and  3.3.5


                 det(B)  =  det(S) det(A)  det(S)-  1  =  det(A).
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