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There seems to be difference between reference frame and coordinate system. See https://en.wikipedia.org/wiki/Frame_of_reference#Definition . A reference frame is something has physical meaning and is related to physical laws, whereas coordinate system (chart) is purely mathematical tools that convert reference frame to numbers. The difference between reference frame and coordinate system is analogous to the difference between vector and number tuple, or linear map and matrix. Is this understanding correct?

I also heard that a reference frame is an ordered set of basis vector (on the tangent space if it's GR). Is this correct? If that's the case, is a non-inertial frame also a set of basis vector?