# Difference between coordinate and vector component transformations

So I have read that components of vectors transform linearly between frames like $$A'^\mu = \frac{\partial x'^\mu}{\partial x^\nu} A^\nu .$$

I have also read that coordinates in general do not transform linearly between coordinate systems, however their differentials do:

$$dx'^\mu = \frac{\partial x'^\mu}{\partial x^\nu} dx^\nu .$$

What I cannot grasp is how on earth can vector components transform between coordinate systems linearly when the coordinates do not? What if the vector in question is a position vector, then the components are the coordinates and then we come to a contradiction? At this point I become confused.

Also, in deriving the vector component transformation, I have always seen the example of transforming from a Cartesian $x$,$y$ plane to another one rotated at an angle $\theta$ relative to the first. Well in this example the components of the vector transform linearly and I can completely see this, but so do the coordinates!

Any help would be much appreciated!

• What do you mean by coordinates do not transform linearly? The transformation between spacetime coordinates is just a Lorentz transformation $\Lambda_{\mu}^{\nu}$. – InertialObserver Jan 2 '17 at 19:42
• Transforming from Cartesian to spherical polar is not linear – Matt0410 Jan 2 '17 at 19:44
• Oh, so you are saying the transformation between different coordinates in that sense, not in the sense of "different reference frame". – InertialObserver Jan 2 '17 at 19:45
• Do you mind my asking where you heard that components of a vector transform linearly? That is not quite correct. – InertialObserver Jan 2 '17 at 20:01
• Well the first equation above is in all textbooks I have read on General Relativity. Doesn't that imply a linear sum of the unprimed components? – Matt0410 Jan 2 '17 at 20:11