The vectors $v_1,v_2,....v_k$ in the vector space $R^n$are eigenvectors for the mateix A. Corresponding to the eigenvalues $\lambda_1,\lambda_2,...\lambda_k$(not necessary distinct). If these vectors are linearly independent and $\lambda_{k+1}$ is distinct from all of the other $\lambda 's$, prove that ${v_1,v_2,....v_k, v_{k+1}}$ is inearly indeoendent for any eigenvector $v_{k+1}$ corresponding to $\lambda_{k+1}$.