Suppose $V={p(x) : p(x) text{ is a polynomial so that its degree is less or equal to 2}}$ and $W={p(x) : p(x) text{ is a polynomial so that its degree is less or equal to 1}}$ are two vector spaces. If $D: V to W$ is a linear transformation given by $D(p(x))=p'(x)$ (the derivative). Find the matrix D related to the bases as stated in: $$D_{{1,x,x^2}}^{{1,x}}$$

My proposal for the matrix is begin{equation*}

A =

begin{pmatrix}

0 & 1 & 0 \

0 & 0 & 2

end{pmatrix}

end{equation*}

for which I define it as a two row, because of the matrix multiplication as the outcome will have two rows too, also that $D(x^k)=kx^{k-1} $. Nevertheless, I am not a hundred percent sure this matrix works or if there is another argument that I still have to show so that this proof is complete, because when proving the general case with the standard basis for polynomials, the same base is used. Link to what I am refering to in the last line: Derivative matrix for n-th degree polynomial