I understand that if $v_1,..,v_r$ are the eigenvectors that correspond to distinct eigenvalues then they are linearly independent (*)
However what if I have say two linearly independent eigenvectors corresponding to one eigenvalue and an eigenvector corresponding to another, with $A$ a $3$x$3$ matrix and $Av=\lambda v$. Are these three eigenvectors linearly independent? Does this follow from (*)?