Commutativity of Operator in Polynomial

Question

I was studying the Following lemma attached in Image given.enter image description here

I know that polynomial is the commutative ring.B ut I am not able to convince my self if we write the product of linear polynomial of the operator is also commutative.
that is $f(T)T=Tf(T)$ WHy?
Any Help will be appereciated

Answer 1

First way to see it: we know that $f(x)g(x) =g(x)f(x)$ for all $f,g\in\mathbb{F}[x]$, thus $f(T)g(T) = g(T)f(T)$. In particular, one can take $g(x) = x$.

Second way: prove that $T^nT = T T^n$ by induction. Writing $f(x)=\sum_i a_ix^i$, it follows that $$f(T)T = \sum_i a_iT^iT = \sum_i a_iTT^i = T\sum_i a_iT^i = Tf(T),$$ as desired.

The idea is that the map $\phi_T\colon\mathbb{F}[x]\to L(V)$, $\phi_T(f) = f(T)$, is a ring homomorphism.