Let $\sigma $ be an invertible linear transformation on linear space $V$. And $W$ is the invariant subsapce of $\sigma$. Prove $W$ is also an invariant subspace of $\sigma^{-1}$.
My try: I was trying to prove $\sigma $ is surjective by $\sigma $ is invertible. But I forgot that $V$ may be infinite. Then I really don't know how to do the steps. Any hints would be helpful.