What are the differences between cylindrical and cartesian coordinates?

[aaaa202]

Consider the attached picture, where they express the unit vectors in cartesian coordinates with the unit vectors in a cylindrical coordinate system:
The questions might be a bit loose, but try to get what I mean and answer as well as you can please :)

1) I find the expression for i, j and k a bit ambigious. Because won't the unit vectors of the spherical coordinate system always be expressed through the cartesian unit vectors like the first lines imply? Can you give me a physical example to help me understand what happens here.

2) To me it is really weird that you say you change basis when seemingly there is some generalized cartesian coordinate frame behind all of it - by this I mean the fact that the unit vectors of the spherical and cylindrical coordinates are themselves expressed in cartesian frames. It's not like they are defined as (ρ,0,0), (0,z,0) and (0,0,\varphi) - Is the reason why you do this that the angle doesn't behave linearly?

 

[fresh_42]

I'm not sure I understood this correctly. Cylindrical and Cartesian coordinates are not transformed by a linear transformation. Their usage depends on the case. Here's a list of common coordinate systems:
https://en.wikipedia.org/wiki/List_of_common_coordinate_transformations