I am a physics student. My mathematical background is quite weak. I just want to know the similarities (if there are any) or differences between coordinate transformation of two kinds :
- Rotation of coordinate (and hence new transformed coordinate system) or translation and so on. (Most of which are symmetry associated)
- Transformation to a different system of coordinates - like Cartesian to cylindrical or spherical to parabolic and so on.
In this regard I would like to know the difference between these two in the linear algebra language. I can see in both cases the inner product is preserved.
Secondly, I am also interested in knowing if we can associate symmetry to the second kind of transformation and hence some generator of transformation, if not what are the cases in which we can't associate such generators to transformation.
I have already posted this question here. However I have not obtained satisfactory responses there.