- #1
mnb96
- 715
- 5
Hello,
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 vector F in that space (F = [tex]f_{1}b_{i}+...+f_{n}b_{n}[/tex])
I'd like to find the components of F in 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!
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 vector F in that space (F = [tex]f_{1}b_{i}+...+f_{n}b_{n}[/tex])
I'd like to find the components of F in 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!