Suppose T is an element of L ( P 3 </msub> ( <mi mathvariant="do

Feinsn 2022-06-26 Answered
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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Answers (1)

Raven Higgins
Answered 2022-06-27 Author has 17 answers
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're looking for.
More generally, it's pretty straightforward to write linear transformations in different bases, α of the domain and β of the range: start by writing the domain'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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