Isomorphic $K[x]$-modules have equal characteristic polynomials

Question

Let $K$ be a field, $V,V'$ be finite dimensional $K$-vector spaces and $A\in \text{End}_K(V),A'\in \text{End}_K(V').$ Regard $V$ (respectively $V'$) as $K[x]$-modules with respect to $A$(resp. $A'$.)

Show that if $V\simeq V'$ as modules, then $A$ and $A'$ have equal characteristic polynomials.

Answer 1

$V$ and $V'$ are isomorphic as $K[x]$-moudle is equivalent to the existence of an isomorphism $f:V\rightarrow V'$ such that $f\circ A=A'\circ f$. Let $(e_1,..,e_n)$ a basis of $V$, $(f(e_1),..,f(e_n))$ is a basis of $V'$. The matrix $M(A,e_1,..,e_n)$ of $A$ in the basis $(e_1,..,e_n)$ is $M(A',f(e'_1),..,f(e'_n))$.