Finding the matrix of a linear transformation with respect to bases

Consider the linear trasnformation $$L:\mathbb{P^3}\rightarrow \mathbb{P^2}$$ given by the formula $$L(p(t))=p'(t)-tp''(t)$$ Find the matrix of the linear transformation $$L$$ with respect to the bases $$S=\{1+t^3,t+t^2,t^2−t^3,t^3\}$$ and $$T=\{1,t,t^2\}$$. Here's what I did to attempt the problem. I found the transformation of the standard bases to be :

$$L(e_1)=\begin{bmatrix}0\\0\\0\\0\end{bmatrix}$$

$$L(e_2)=\begin{bmatrix}0\\0\\0\\0\end{bmatrix}$$

$$L(e_3)=\begin{bmatrix}0\\0\\0\\0\end{bmatrix}$$

$$L(e_4)=\begin{bmatrix}0\\-3\\0\\0\end{bmatrix}$$

So I know that the matrix $$A$$ representing the linear transformation is :

$$A=\begin{bmatrix}0&0&0&0\\0&0&0&-3\\0&0&0&0\\0&0&0&0\end{bmatrix}$$

I don't know what to do after this step. How do I find the matrix with respect to both bases $$S$$ and $$T$$? I apologize if I did something horrible wrong or if the answer is obvious!

• Did you mean $p''(t)$ when you wrote $t''(p)$? Oct 15 '19 at 17:12
• and $L(e_1)$ when you wrote $T(e_1)$? Oct 15 '19 at 17:13
• That's the standard basis for $\Bbb R^4$; what's $\Bbb P^3$? Oct 15 '19 at 17:25
• Oh, I get it now, $T$ is the basis for the codomain, so find what $L$ does to each element of $S$ and express the results as linear combinations of elements of $T$ Oct 15 '19 at 17:28
• Do you know what $L(s)$ is for each $s\in S$? Oct 15 '19 at 17:36

$$S=\{1+t^3,t+t^2,t^2−t^3,t^3\}$$ is a basis of $$\Bbb P^3$$, the domain of $$L$$,
and $$T=\{1,t,t^2\}$$ is a basis of $$\Bbb P^2$$, the codomain of $$L$$.
$$L$$ maps the elements of $$S$$ to $$-3t^2, 1, 3t^2, -3t^2$$, respectively,
so $$A=\begin{bmatrix}0&1&0&0\\0&0&0&0\\-3&0&3&-3\end{bmatrix}.$$
Note that $$A$$ has the proper size for mapping a $$4$$-dimensional space to a $$3$$-dimensional space.