Diagonalizable Operators: An Operational Extension Let $T$ be a diagonalizable operator on a vector space $V$. Prove that the operator
$$a_nT^n + a_{n-1}T^{n-1}+\cdots+a_1T+a_0 Id_V$$
 on $V$ is also diagonalizable for any scalars $a_1, a_1,\dots,a_n$.



First off, what is $Id_V$? I've never seen this before.
Hint:

$$\exists\,P\;\,\text{an invertible operator}\,\;s.t.\;\, P^{-1}TP=D\;\,\text{is diagonal}\implies \forall\,n\in\Bbb N:$$

$$P^{-1}T^nP=\left(P^{-1}TP\right)^n=D^n\;\;\text{is diagonal,  and from here the claim follows}$$
I found this in Michael Artin's Algebra, and I feel it will be of use here: