Linear independence of real and imaginary parts of complex eigenvector

Observe that if $Av=\lambda v$, then $\overline{Av} = \overline{\lambda v} = \overline\lambda \overline v,$ but since $A$ is real, $\overline A=A$, so $\overline v$ is an eigenvector of $A$ with eigenvalue $\overline\lambda$. This implies that if $\lambda$ is complex, then $v$ and $\overline v$ are linearly independent. If we have $$c_1\Re(v)+c_2\Im(v) = \frac {c_1}2(v+\overline v) - i\frac{c_2}2(v-\overline v) = {c_1-ic_2\over2}v+{c_1+ic_2\over2}\overline v = 0,$$ then we must have $c_1=c_2=0$ because $v$ and $\overline v$ are linearly independent.