1. ## Kernel and Image

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).

I need to find a basis for each of the kernel ker(T) of T, and the image (or range) Im(T) of T (T is a linear operator on V).

Any ideas?

2. Originally Posted by PvtBillPilgrim
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).

I need to find a basis for each of the kernel ker(T) of T, and the image (or range) Im(T) of T (T is a linear operator on V).

Any ideas?
After you find the standard matrix for the linear transformation does it not make sense to say that the kernel of the map of the nullspace of the standard matrix and the range of the map is the coloum space of the standard matrix? However, what that standard matrix is, I did not find it.