Linear mapping/Trigonalizable/Characterization with characteristic polynomial/Fact

Characterization of trigonalizable mappings

Let ${}K$ denote a field, and let ${}V$ denote a finite-dimensional vector space. Let

$\varphi \colon V\longrightarrow V$

denote a linear mapping. Then the following statements are equivalent.

1. ${}\varphi$ is trigonalizable.
2. The characteristic polynomial ${}\chi _{\varphi }$ has a factorization into linear factors.

If ${}\varphi$ is trigonalizable and is described by the matrix ${}M$ with respect to some basis, then there exists an invertible matrix ${}B\in \operatorname {Mat} _{n\times n}(K)$ such that ${}BMB^{-1}$ is an upper triangular matrix.

Write a proof