Let $V$ be a finite dimensional $K$-vector space, let $$\alpha ∈ End_k(V)$$ be a $K$-endomorphism of $V$ , and let $S$ be a $K$-basis of $V$ .
Prove that for every polynomial $f \in K[X]$ one has the equality: $$ M_{S,S}(f(\alpha)) = f(M_{S,S}(\alpha))$$
I think I need to fix a polynomial and then work on the given equality by focusing on the input and output of matrix $M$ which is $S$. Could you please give me a hint on how to prove it ? I really do not know from where to begin?
Well, we can start by proving then using the following: \begin{align*} M_{s, s}(\alpha + \beta) &= M_{s, s}(\alpha) + M_{s, s}(\beta) \\ M_{s, s}(\alpha\beta) &= M_{s, s}(\alpha) M_{s, s}(\beta) \\ M_{s, s}(k \alpha) &= k M_{s, s}(\alpha), \end{align*} for any endomorphisms $\alpha$ and $\beta$, and any scalar $k$. You can use induction and the second equality to show that $M_{s, s}(\alpha^n) = (M_{s, s}(\alpha))^n$ for any $n$. You can then use the third equality to show that the result holds for any monomial function (i.e. a function of the form $f(x) = kx^n$). Finally, you can use induction a second time to prove that the result works for sums of monomials, a.k.a. polynomials!