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say we have a non orthonormal basis, what do we do to get the correct result?

project it back to a orthonormal basis, then do the computation?


In what situations is it most likely that it is not as convenient to have unit length or mutually orthogonal vectors? (since the first bullet is opened by 'most often')


To ask a similar question, why would you ever want a basis that is not orthonormal?


Can you show an example of projecting the same vector onto different basis (pref. one orthonormal and one NOT orthonormal)?


In which cases (if any) would a non-orthonormal basis be wanted or needed and why?


I feel in some cases, I need to use polar coordinates. This is surely a non-orthonormal basis and it makes me confused.


Could you do an example where you take a vector in a non orthonormal basis and show how the norm is no longer the same? When I write one as a linear combination of my basis it seems I am always using the basis of (1,0) and (0,1).


Is there any use in choosing an non-orthonormal basis?


Would we ever need to convert from an orthonormal basis to a non-orthonormal basis?