Endomorphism/Eigenvectors/Linearly independent/Fact

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

$\varphi \colon V\longrightarrow V$

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

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

Proof, Write another proof