I have two functions $f_a(x) = y_a$ and $f_t(x) = y_t$, where the dimension of the domain is larger than range. Finally, I want to compute the linearised mapping from $y_t$ to $y_a$ via ${}^t J_a = \frac{\partial y_a}{\partial y_t}$.
I can not compute this directly but can compute the Jacobians $J_a= \frac{\partial y_a}{\partial x}$ and $J_t = \frac{\partial y_t}{\partial x}$ numerically. Furthermore, I know, that $\mathrm{kernel}(J_t^TJ_t) = \mathrm{kernel}(J_a^TJ_a)$.
Can I do better then the pseudo inverse $J_a J_t^{\dagger} = \frac{\partial y_a}{\partial x} \frac{\partial x}{\partial y_t}$ to compute $\frac{\partial y_a}{\partial y_t}$?
What conclusions can I draw from the information of identical null spaces?