Page 128 - 35Linear Algebra
P. 128

128                                                                                      Matrices


                            By linearity this specifies the action of L on any vector from V as

                                                                           
                                                       1        0                 0
                                                       1        1    = (c 1 + c 2 )  1  .
                                               L c 1     + c 2              
                                                 
                                                       0        1                 0
                            We had trouble expressing this linear operator as a matrix. Lets take input basis
                                                           
                                                            1     0
                                                                  1
                                                               ,
                                                   B =            =: (b 1 , b 2 ) ,
                                                            1
                                                            0     1
                            and output basis
                                                               
                                                             1      0     0
                                                                 ,
                                                                       ,
                                                                    1
                                                     E =                .
                                                             0
                                                                          0
                                                             0      0     1
                            Then
                                                        Lb 1 = 0e 1 + 1e 2 + 0e 3 ,
                                                        Lb 2 = 0e 1 + 1e 2 + 0e 3 ,
                            or

                                                                                            
                                                          0               0                  0 0

                                                          1
                                  Lb 1 , Lb 2 ) = (e 1 , e 2 , e 3 )    , (e 1 , e 2 , e 3 )     = (e 1 , e 2 , e 3 )   1 1   .
                                                                          1
                                                          0               0                  0 0
                            The matrix on the right is the matrix of L in these bases. More succinctly we could
                            write
                                                                         
                                                                          0

                                                          x
                                                       L       = (x + y)   
                                                                          1
                                                          y
                                                             B            0
                                                                             E
                                                                      
                                                                   0 0
                            and thus see that L acts like the matrix   1 1 .
                                                                       
                                                                   0 0
                               Hence
                                                                          
                                                                  0 0

                                                       x                  x
                                                    L       =   1 1          ;
                                                       y                  y
                                                          B       0 0
                                                                               E
                            given input and output bases, the linear operator is now encoded by a matrix.
                            This is the general rule for this chapter:
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