Problem: let T be a linear mapping on a finite-dimensional vector space V, and suppose there exists an ordered basis $\beta$ for V such that $[T]_\beta$ is an upper triangular matrix. Prove that the characteristic polynomial for T splits.
Proof: the characteristic polynomial is given by $f(t)=| A-\lambda I$|, since $A=[T]_\beta$ is upper triangular, we can write it as $f(t)=\Pi_{i=1}^{n}([T]_{\beta_{ii}}-t)$. Hence the characteristic polynomial splits.
Question: Anything failed to consider?