Prove that $d$ is a linear transformation

Question

Let $n \in \mathbb N$ and let $d:\,R_n[X]\to R_n[X]$ be the derivative linear transformation, s.t. $d(P)=P'$.

I know that map f : V → W is said to be linear, if for any u and v vectors in V , for any real number λ, we have

(1) f(u + v) = f(u) + f(v),

(2) f(λu) = λf(u).

How can I prove exactly that the operation of taking derivatives satisfies the sum rule, and the condition with taking out a constant?

Also, how can I find the characteristic polynomial and the minimal polynomial of $d$? I have found the associated matrix $A$ but I have no idea how to do so here:

\begin{bmatrix} 0 & 1 & \dots & 0 \\ 0 & 0 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & n-1 \\ 0 & 0 & \dots & 0 \end{bmatrix}

Answer 1

Hint

For the linearity, what could be $d(\lambda P+Q)$ where $P,Q$ are polynomial and $\lambda \in\mathbb R$ ?

For the caracteristic/minimal polynomial, remark that $d^{n+1}(X^m)=0$ for all $m\leq n$. I recall that $d^n=\underbrace{d\circ ...\circ d}_{n\ times}$.