Suppose T is an element of L ( P 3 ( R ) , P 2 ( R ) ) is the differentiation map defined by T p = p ′ . Find a basis of P 3 ( R ) and a basis of P 2 ( R ) such that the matrix of T with respect to these basis is ( 1 0 0 0 0 1 0 0 0 0 1 0 )

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Suppose   T is an element of   L  (      P    3    (  R  )  ,      P    2    (  R  )  ) is the differentiation map defined by   T  p  =      p    ′  . Find a basis of       P    3    (  R  ) and a basis of       P    2    (  R  ) such that the matrix of T with respect to these basis is       (                            1                          0                          0                          0                                      0                          1                          0                          0                                      0                          0                          1                          0                      )

Raven Higgins · Accepted Answer

With       e    1    =      1    3        x    3    ,      e    2    =      1    2        x    2    ,      e    3    =  x  ,      e    4    =  1 represented by vectors      (                            1                  /          3                                      0                                      0                                      0                      )    ,      (                            0                                      1                  /          2                                      0                                      0                      )    ,      (                            0                                      0                                      1                                      0                      )    ,      (                            0                                      0                                      0                                      1                      )  relative to the standard basis, and a basis for the target space as usual just       x    2    ,  x  ,  1 represented by the usual basis vectors.The map in this basis is exactly what you&#039;re looking for.More generally, it&#039;s pretty straightforward to write linear transformations in different bases,   α of the domain and   β of the range: start by writing the domain&#039;s basis in terms of the standard basis and getting a matrix       T    α   which changes the standard basis into the basis,   α. Next find a matrix       T    β   which changes the standard basis on the domain space into the basis you want. Finally find a matrix       T    0   which does the job on the standard basis.Then your map is   T  =      T    β        T    0        T    α          −      1      ,  which you can verify by testing it on a basis,       T    α          −      1       takes an alpha basis vector, turns it into a standard one,       T    0   does the work of the transformation, and       T    β   writes it in the alternate basis.

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