Linear Transformations, Kernel and Image of Matrices

"Let P2 be the vector space over R of polynomials of degree at most 2, and let F: P2 -> P2 be the function defined by F(p)(x) = p'(x) + 3(p)''(x). Prove that F is a linear transformation. Find the matrix F with respect to the basis {1,x,x^2} of P2. Find bases for Ker(F) and Im(F)."

Right, I can do the first part, showing its a linear transformation, am I right in thinking that to find the matrix I find F(1), F(x) and F(x^2)?

This gives:

( 0 1 6 )

( 0 0 2 )

( 0 0 0 )

Is this correct? Could anybody please tell me how to find the bases for ker(F) and im(F) from here?

Thanks