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[tex] \langle b_i|O|b_j\rangle [/tex]

Where the "b's" are vector of the basis set. Does this rule work if the basis is not orthonormal? Because I was checking this with regular linear algebra (in R

^{3}) (finding matrix elements of linear transformations) and this only seems to work with the canonical basis. The same goes for the rule that allows you to find the coefficients of the expansion of a vector in a given basis:

[tex] |\psi\rangle =\[

\sum_{i=1}^{\infty} c_i |\psi\rangle

\] [/tex]

with

[tex] c_i = \langle b_i|\psi\rangle [/tex]