Linear transformations and eigenvalues Let $T: \mathbb C^n \rightarrow \mathbb C^n$ be linear. Let $\beta$ and $\gamma$ be any two ordered bases. Prove that the eigenvalues of $[T]_\beta$ and $[T]_\gamma$ are the same. 
Can anyone provide tips/hints in the right direction? I'm struggling as I try to understand this intuitively....thank you

The eigenvalues are a property of the operator $T$. Recall that $\lambda$ is an eigenvalue of $[T] $ if $0=\det([T]-\lambda\,I)=\det([T-\lambda\,I]) $. This happens precisely when the operator $T-\lambda\, I$ is not invertible, i.e. there exists $v\in\mathbb C$ with $Tv=\lambda v$. This shows that the eigenvalues of $[T]$ (in any basis) are precisely the eigenvalues of $T$.
