Let $P_3(\mathbb{R})$ be a real vector space with the basis $V = (1,x,x^2)$.
Find the kernel and the image of the image of the linear operator: $$D: P_3(\mathbb{R}) \to P_3(\mathbb{R})$$ $$p(x) \to p'(x)$$
With respect to the basis $V$ we know that: $$D(1) = 0 + 0x + 0x^2$$ $$ D(x) = 1 + 0x + 0x^2$$ $$ D(x^2) = 0 +2x + 0x^2$$
So the matrix representation would be: $$ _V[D]_V = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 2 \\ 0 & 0 & 0 \end{pmatrix}$$
Thus we know that $x_1 =$ free variable, $x_2 = 0$ and $x_3 = 0$. The null space of the matrix representation is $\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$
The image of the matrix representation is $(\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix},\begin{pmatrix} 0 \\ 2 \\ 0 \end{pmatrix})$
My question is, how do I find the kernel and the image of the linear operator, when I know the Null space and the image of the corresponding matrix representation?