Show that the only invariant lines for the linear system $x' = Ax$ with $x \in \Bbb {R^2}$ are the lines $ax_1+bx_2=0$ where $v=(-b,a)^T$ is an eigenvector of A.
I understand that if this line (let's call it L) is invariant for the system, it means that for all $x_o \in L$, $e^{At}x_o \in L$ for all real t. What I do not understand is really how to go about proving this. Any assistance would be welcomed.