I'm trying to self-study Linear Algebra from Linear Algebra Done Wrong, but the book doesn't have solution manual so my question might be extremely easy, apologize in advance:
The question is for the linear transformation below find its matrix:
$T:\mathbb{P_n}\rightarrow\mathbb{P_n};Tf(t)=2f(t)+3f^{'}(t)-4f^{''}(t)$ with respect to the standard basis $1,t,t^2,...,t^n$.
Firstly, I defined $\mathbf{e}_0:=1;\mathbf{e}_1:=t;...;\mathbf{e}_n:=t^n$ for the polynomial, and then I reduced the transformation to $\sum_{k=1}^{n}a_kT(\mathbf{e}_k)$ and then I evaluated $T(\mathbf{e}_k)$ for all $k$. For example $T(\mathbf{e}_0)=\begin{pmatrix} 2\\ 0\\ 0\\ \vdots\\ 0\\ 0\\ 0\\ \end{pmatrix}$ and so on until $T(\mathbf{e}_n)=\begin{pmatrix} 0\\ 0\\ 0\\ \vdots\\ -4n(n-1)\\ 3n\\ 2\\ \end{pmatrix}$. After that, I returned to the $\sum_{k=1}^{n}a_kT(\mathbf{e}_k)$, factored out the $a_k$ and got the matrix:$\begin{pmatrix} 2 & 3 & -8 & \cdots & 0\\ 0 & 2 & 6 & \cdots & 0\\ 0 & 0 & 2 & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & 0 & \cdots & -4n(n-1)\\ 0 & 0 & 0 & \cdots & 3n\\ 0 & 0 & 0 & \cdots & 2 \end{pmatrix}$
Now, Is my reasoning correct? If it isn't, where's the problem and how one could solve this solution and if it is, Is this the usual way to derive the matrix or there's a simpler way to do that? Is the matrix correct after all? Any help would be appreciated!