"Let $B=P^{-1} A P$ for some invertible matrix $P \in mat(n;F)$ Prove that for any $\lambda \in F, E(\lambda,B) \rightarrow E(\lambda,A), \vec{w} \rightarrow P\vec{w}$ is an isomorphism of F vector spaces."
Playing around with definitions I think this acting as a change of basis for eigenvectors. I was trying to prove there is an inverse to show the bijection, the best I have is:
$AP\vec{w} = \lambda P\vec{w}$
$P^{-1}AP\vec{w} = \lambda P^{-1} P \vec{w}$
$B\vec{w} = \lambda \vec{w}$
which although I think implies an inverse must exist as it sends $P \vec{w} \rightarrow \vec{w}$ I can't see this is good enough as a proof.
Any feedback/suggestions would be appreciated, thanks
Reading your proof bottom-up is a proof that your function $f$ is well-defined, so it's the first step.
To show that it is an isomorphism, you have to show that it is linear (trivial), injective (show $f(w)=0\Rightarrow w=0$) and surjective (everey $\lambda$-eigenvector of $A$ is reached by $f$).