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176              Chapter  Six:  Linear  TYansformationns


             We shall  encounter  other  common  properties  of similar  matri-
             ces  in  Chapter  Eight.


             Exercises   6.2

             1.  Which   of  the  following  functions  are  linear  transforma-
             tions?
                  (a)  Ti  :  R 3  —>  R  where  Ti([2:1X2X3])  =  \Jx\  + x\  + x§;
                                                                     T
                  (b)  T 2  : M m, n{F)  -  M n , m (F)  where  T 2(A)  =  A ;
                  (c)  T 3  : M n(F)  ->• F  where  T 3 (^)  =  det(4).
             2.  If  T  is a  linear  transformation,  prove that  T(—v)  =  —T(y)
             for  all  vectors  v.
             3.  Let  I  be  a  fixed  line  in  the  xy-plane  passing  through  the
             origin  O.  If  P  is  any  point  in  the  plane,  denote  by  P'  the
             mirror  image  of  P  in  the  line  /.  Prove  that  the  assignment
                                                                 2
             OP   —>  OP'   determines  a  linear  operator  on  R .  (This  is
             called  reflection  in  the  line  I).
             4.  A  linear  transformation  T  : R  —>  R  is  defined  by


                         Xl
                        ( \              Xi   —  X2   —  %3  ~  £4
                     T{                 2xi   +  x 2  -  X3
                          X3
                                                 %2   -  %3  +  %4
                         \X4  /

             Find the matrix that  represents  T  with respect to the  standard
                                 3
             bases  of  R 4  and  R .
             5.  A  function  T  :  P^(R)  —>• P^(R)  is  defined  by  the  rule
                                    /
             T(f)  =  xf"  —  2xf  + .  Show that  T  is  a  linear  operator  and
             find the matrix that  represents  T  with  respect  to the  standard
             basis  of  P4.(R).
             6.  Find  the  matrix  which  represents  the  reflection  in  Exercise
                                                                2
             3 with  respect  to the  standard  ordered  basis  of  R ,  given  that
             the  angle  between  the  positive  x-direction  and  the  line  I is  4>.
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