I have to solve the following problem:
Let V be the R-space of real polynomials of degree at most 3.
For p(X) in V, define
Tp(X)=1/6(19-6x+7x^2+32x^3-15x^4-6x^5)p'''(X)+(6-7x-9x^2+5x^3+x^4)p''(X)+(7-5x^2+2x^3)p'(X)+(10-6x^2)p(X).
Find [T]s where S is the standard ordered basis {1,x,x^2,x^3}.
Find [T]b where B is the non-standard ordered basis {2-12x-x^2+4x^3, 2-x^2, 3-x^2, -5x+2x^3}.
How do I work these out?
You find the transformations of the basis vectors.
Then you find their respective coordinate vectors relative to the second base.
Then the matrix obtained by the cololum coordinate vectors is the standard matrix for the linear transformation.
(That was the same thing I wrote only in nota
tion)