Prove that T is a linear transformation

Hello,

T: P2(R) -> P3(R) defined by T(f(x)) = xf(x) + f`(x).

Let A,B,C,a,b,c be elements of R and x,y elements of P2, where f(x) = (a + bx + cx^2) and f(y) = (A + By + Cy^2)

Can't I write this as a matrix vector product and prove it to be linear that way? I can also calculate the rank and null this way, right?

T(f(x)) = x(a + bx + cx^2) + b + 2x

= b + 2ax + bx^2 + cx^3

is this right so far? I really want to use the matrix vector product method though, but I'm having a hard time making it into a matrix, any advice?