I am struggling to show that two exponential curves in $\mathbb{R}^n$ do not overlap except finite distinct points. Could anyone help me?
Let $A\in\mathbb{R}^{n\times n}$ and $a_x,a_y,b\in\mathbb{R}^n$. Consider two curves $f,g: \mathbb{R}\to\mathbb{R^n}$, which are defined as $$ f(x)=e^{Ax}a_x,\quad x\in[x_m,x_M]\subset\mathbb{R},\\ g(y)=e^{Ay}a_y+b,\quad y\in[y_m,y_M]\subset\mathbb{R}. $$ Show that
two curves $f(x)$ and $g(y)$ do not have infinite crossing points except the case that $b={\bf 0}$ and $\exists z\in\mathbb{R}$ s.t. $a_x=e^{Az}a_y$.
Or, show that
there exists an open interval $(x_1,x_2)$, $x_m\leq x_1< x_2\leq x_M$ such that $\forall x\in(x_1,x_2)$, $\exists y\in[y_m,y_M]$ and $f(x)=g(y)$ only if $b={\bf 0}$ and $\exists z\in\mathbb{R}$ s.t. $a_x=e^{Az}a_y$.
Once the second statement has been shown, we can show that the first one since $x$ and $y$ are bounded.
Intuitively, the statements are true: Curves $e^{Ax}a_x$ and $e^{Ay}a_y$ do not have intersection if $a_x\neq e^{Az}a_y,\forall z\in\mathbb{R}$. Thus, $g(y)=e^{Ay}a_y+b$ may across $e^{Ax}a_x=f(x)$ at some distinct points, but might not overlap.