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If there is no geometric meaning if the coordinates are not from an orthonormal basis, then where do we uses inner products for non-orthonormal basis? Or do we rather convert the coordinates into an orthonormal basis and then do the inner product to get a geometric meaning out of it?


It sounds like most things play nice with an orthonormal basis. Is it a good idea then to wrap all operations into a process that first converts vectors to an orthonormal basis and then applies said operation?


Is there a good visualization/intuition for why the dot product is meaningless without an orthonormal basis?