Proving 2-Dimensional Maps Are Topologically Conjugate

Question

I am having some difficulty proving the fact that the maps

$$F(v)=\left(\begin{matrix} 1/2&0\\ 0&1/2 \end{matrix}\right)v$$

and

$$G(v)=\left(\begin{matrix} 1/4&0\\ 0&1/8 \end{matrix}\right)v$$

are topologically conjugate as maps on $\mathbb{R}^2$.

I am allowed to use the fact that similar matrices are topologically conjugate and that maps on $\mathbb{R}$ defined by $f(x)=ax$ and $g(x)=bx$ with $0<a<b<1$ are topologically conjugate. Any help is appreciated.

Answer 1

Hint: Rewrite the maps as $$ F(v) = (f_1(v_1),f_2(v_2)) = \left(\frac 12 v_1,\frac 12 v_2\right)\\ G(v) = (g_1(v_1),g_2(v_2)) = \left(\frac 14 v_1,\frac 18 v_2\right). $$ We can see that $f_i$ is topologically conjuagte to $g_i$ for $i=1,2$. How does this allow us to answer the question?