The vectors $\displaystyle v_1, v_2, ..., v_k$ in the vector space $\displaystyle R^n$ are eigenvectors for the matrix $\displaystyle A$ corresponding to the eigenvalues $\displaystyle \lambda_1, \lambda_2, ..., \lambda_k$ (not necessarily distinct).

If these vectors are linearly independent and $\displaystyle \lambda_{k+1}$ is distinct from all of the other $\displaystyle \lambda$'s, prove that $\displaystyle \{ v_1, v_2, ..., v_k, v_{k+1} \}$ is linearly independent for any eigenvector $\displaystyle v_{k+1}$ corresponding to $\displaystyle \lambda_{k+1}$.

(Hint: if not, we have $\displaystyle v_{k+1}=c_1v_1+c_2v_2+...+c_kv_k$ for some constants and $\displaystyle Av_{k+1}$ can easily be written in two ways.)

Please show your work.