Let $T:V\rightarrow V$ be Linear Transformation and $f(t)$ be any Polynomial, then $\ker f(T)$ is invariant under T.

Question

Let $T:V\rightarrow V$ be Linear Transformation and $f(t)$ be any Polynomial, then $\ker f(T)$ is invariant under T.

I tried to prove. I am stuck with the last step.

Let $f(x)=a_{0}+a_{1}x+\cdots+a_{n}x^{n}$

$x\in \ker f(T)$

$\implies f(T)(x)=0$

We need to prove that $T(x)\in \ker f(T)$

$f(T)(T(x))=a_{0}T(x)+a_{1}T(T(x))+\cdots+a_{n}T^{n}(T(x))$

$...$

How to prove that this quantity is zero?

Answer 1

This quantity is$$T\bigl(a_0x+a_1T(x)+\cdots+a_nT^n(x)\bigr)=T(0)=0.$$