eigenspaces/diagonalizability

Hey y'all I got a question regarding eigenspaces which is stumping me. Heres the problem, any tips or help would be greatly appreciated.

Let T be an invertible linear operator on a finite-dimensional vector space V.

Prove that the eigenspace of $\displaystyle T$ corresponding to $\displaystyle \lambda$ is the same as the eigenspace of $\displaystyle T^{-1}$ corresponding to $\displaystyle \lambda ^{-1}$

Also prove that if $\displaystyle T$ is diagonalizable, then $\displaystyle T^{-1}$ is diagonalizable.

Thanks!