If $T$ is an invertible linear transformation and $\vec{v}$ is an eigenvector of $T$, then $\vec{v}$ is an eigenvector of $T^{-1}$ I saw there is a proof for invertible matrices, but I don't know how to put this mathematically for a transformation.
How do I prove an invertible linear transformation has the same eigenvectors as its inverse?

Observe that for an invertible matrix $A$ with eigenvector $\mathbf v$ and corresponding eigenvalue $\lambda\neq 0$, you have that 
\begin{align*}
\mathbf v &= I \mathbf v\\&=A^{-1}A \mathbf v\\&=\lambda A^{-1} \mathbf v
\end{align*}
Hence $A^{-1} \mathbf v = \lambda^{-1}\mathbf v$
