Eigenvectors with distinct eigenvalues are linearly independent.

Theorem: Eigenvectors corresponding to distinct eigenvalues are linearly independent. If $v_1, \ldots, v_k$ are eigenvectors of $A$ with distinct eigenvalues $\lambda_1, \ldots, \lambda_k$, then $\{v_1, \ldots, v_k\}$ is linearly independent. Proof: By induction using the fact that if a linear combination equals zero, applying $A$ and subtracting $\lambda_k$ times the original equation eliminates $v_k$.