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!
 A: In a general curved spacetime, vectors are only defined in the tangent space of individual points, and are by definition part of a vector space. This is the only thing that makes sense since one cannot assign a consistent rule to map vectors at one point to another in a curved spacetime. This is related to the fact that there is a non-trivial holonomy when there is non-zero curvature. Coordinates on the other hand are defined on larger patches of the spacetime. 
General transformations of coordinates could be non-linear, but they would always induce a local transformation on vectors within their own tangent spaces. This would then result in a linear transformation on the vectors.
The example of the Cartesian plane is confusing because it has zero curvature. Try this with some other non-trivial manifold like a sphere.
A: For clarification you have to distinguish between coordinate systems and sets of base vectors. The linearity in transforming the vector components is a consequence of the properties of vector spaces. Coordinate transformations are somewhat different from base changes.
