Linear Independence of Eigenvectors

Theorem: Linear Independence of Eigenvectors

Let $A\in F^{n\times n}$ be a square matrix.

If ${\vec{e}}_1,\cdots ,{\vec{e}}_r$ are eigenvectors which belong to $r$ different eigenvalues ${\lambda }_1,\cdots ,{\lambda }_r$ respectively, then they are linearly independent.

PROOF

TODO