Endomorphism/Eigenvectors/Linearly independent/Fact

Let ${\displaystyle {}K}$ be a field, ${\displaystyle {}V}$ a ${\displaystyle {}K}$-vector space and

${\displaystyle \varphi \colon V\longrightarrow V}$

a linear mapping. Let ${\displaystyle {}v_{1},\ldots ,v_{n}}$ be eigenvectors for (pairwise) different eigenvalues ${\displaystyle {}\lambda _{1},\ldots ,\lambda _{n}\in K}$.

Then ${\displaystyle {}v_{1},\ldots ,v_{n}}$ are linearly independent.