Polynomial of a linear operator

Question

Let $f(x),g(x),a(x),b(x)\in \mathbb{K}[x]$ be such that $a(x)\cdot f(x)+b(x)\cdot g(x)=1$.

$\mathbb{K}$ can be any field. If $\phi\in End_{\mathbb{K}}(V)$. ($V$ a finite-dimensional $\mathbb{K}$-vector space). Why is the following always true:

$a(\phi)\circ f(\phi)+b(\phi)\circ g(\phi)=Id_{V}$.

I don't know if this should be trivial, can you elucidate me. My linear Alg. Prof. used this several times in prooving spectral theorems.

Answer 1

The key is: What does this notation $a(\phi)$ means? How to apply a polynomial to a linear endomorphism $\phi$?

And the answer begins by this: for a constant polynomial $\lambda$, $\lambda(\phi) := \lambda\cdot Id$, and for $x$, $x(\phi):=\phi$. The rest is straightforward..