I would like to ask a question about partial derivatives in the context of Rotations of coordinate systems. Say we have a coordinate system (unprimed) and its rotated version (primed). If the relation between the coordinates is linear (Does it have to be? Please, explain), how could I prove the following equation:
$${\frac{\partial x'_i}{\partial x_j}=\frac{\partial x_j}{\partial x'_i}}$$
I am not sure I have to use the chain rule or I could come up with some simple explanation (basic). But I keep moving in circles. Please if someone could shed some light into this I would appreciate it. Thank you in advance!