Maybe this question will sound silly, but I cannot formulate a complete understanding.
I have observed that the standard examples for change of basis always involves one standard basis. Usually two non-standard basis sets will be given, w.r.t the standard basis, and we are required to calculate the transformation matrix between these two. My question is
What if the standard basis is not unitary? What if we have a very different standard basis?
Will the transformation matrix depend on the choice of the coordinate system chosen for the standard basis? Like Cartesian unitary vs. polar/cylindrical?