In $\mathbb{R}^2$, have the linear systems $X' = MX$ and $Y' = NY$ where $N = Q^{−1}MQ$ for invertible matrix $Q$. Show that the two systems are topologically conjugate and that $Q$ (or $Q^{−1}$) is a homeomorphism.
I need to show that $Q$ will map the solutions of $X'$ to solutions of $Y'$ right? and that $Q$ is bijective, continuous, and $Q^{-1}$ is continuous too.
For this, is it enough to that for $Y(t) = QX(t)$, we will have $QdX(t)/dt = NY$? How can I show this?