Yes, that's exactly right. The "kernel" of a linear transformation is the set (a subspace) of vectors that the linear transformation takes to 0. If T(a+ bx+ cx^2+ dx^3)= a, then a+ bx+ cx^2+ dx^3 is in the kernel of T if and only if a= 0. That is thethreedimensional subspace of all polynomials of the form bx+ cx^2+ dx^3. The kernelis{{P(x) | P(x)= bx+ cx^2+ dx^3}.

Thanks, Deveno. I have no idea now why I wrote "four"!