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I am reading a text in which it is assumed the following to be known:

- a n-dimensional Hilbert Space

- a set of non-orthogonal basis vectors{[tex]b_{i}[/tex]}

- a vectorFin that space (F= [tex]f_{1}b_{i}+...+f_{n}b_{n}[/tex])

I'd like to find the components ofFin the given basis{[tex]b_{i}[/tex]}, and according to the text this is easily done by:

[tex]f_{j}[/tex] = [tex]<F,b_{i}><b_{i},b_{j}>^{-1}[/tex]

where Einstein summation convention has been used on the index 'i'.

I'd really like to know how I could arrive at that formula, but as far as I know, since the system is non-orthogonal the dot-product [tex]<F,b_{i}>[/tex] yields the corresponding coordinate in the dual base system{[tex]B_{i}[/tex]}, and so I was able to arrive only at the following formula:

[tex]f_{j}[/tex] = [tex]<F,b_{i}><B_{i},B_{j}>[/tex]

but can I get rid of the dual-basis vectors in the formula?

Thanks a lot in advance!

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# How to find vector coordinates in non-orthogonal systems?

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