I have a transformation, $T(p(x)) = p(x+1) - p(x) -p'(x)$ on $R[x]_3$ (polynomials of degree 3) and I want the matrix for T. No basis is specified, so I'm assuming the standard, what should I do and how should I think about the problem?
Also, the question asks for an explicit description of the kernel and image of T. I'm not really sure what that means, could you explain it please?
Hint:
In the standard basis, the column vectors are the coordinates of $T(1),\, T(x),\, T(x^2),\, T(x^3)$ (in that order).