I get confused about the difference in accessibility between linear and nonlinear systems.
For nonlinear systems with the form \begin{equation} \dot{x}(t)=f(x)+\sum_{i=1}^m g_i(x)u(t). \end{equation} The book (A. Isidori, Nonlinear Control Systems, Springer-Verlag, 1995.), used the following distribution to study the accessibility (reachability) from initial state $x_0$: $$\mathcal{R}_{nonlinear}=\{f,g_1,\dots,g_m\vert \mathrm{span}\{f,g_1,\dots,g_m\}\},$$ where it means the smallest distribution invariant under $f,g_1,\dots,g_m$ and contains $f,g_1,\dots,g_m.$
When $f(x)=Ax, g_i(x)=b_i$, it turns to linear systems, which can be written as \begin{equation} \dot{x}=Ax+Bu. \end{equation} Using the above definition, we have $$\mathcal{R}^*_{nonlinear}=\{Ax\vert \mathrm{span}\{Ax_0,b_1,\dots,b_m\}\}. $$
However, the classical fact about the reachability of linear systems is that $$\mathcal{R}_{linear}=\{Ax\vert\mathrm{span}\{ b_1,\dots,b_m\}\}.$$
I wonder why $\mathcal{R}_{linear}$ doesn't have to contain $Ax_0$.
Thank you for your explanations and help.