Solving for variable inside summations

[sebasbri]

Hi,

I have an equation:

$\frac{q-\bar{q}}{w_{i}^{2}+w'_{j}^2}=\sum^{n}_{i=0}\sum^{m}_{j=1}F_{0}\cdot a_{ij}\cdot cos(w_{i}\cdot x)\cdot cos(w'_{j}\cdot y)$

I'm trying to solve for $a_{ij}$

I have the solution, but I'm not sure how they came up with it. The solution is:

$a_{ij}=\frac{\int^{s}_{0}\int^{w/2}_{0}q-\bar{q}/F_{0}\cdot cos(w_{i}\cdot x)\cdot cos(w'_{j}\cdot y)\cdot dy\cdot dx}{(w_{i}^{2}+w'_{j}^2)\cdot s \cdot w/8}$


Any help would be appreciated. Even if it's just a small push in the right direction.

Cheers

 

[HallsofIvy]

They are using the fact that, if $\{v_i\}$ is an orthonormal basis for an inner product space and $v= \sum a_iv_i$, then $a_i= <v_i, v>$ where < , > is the inner product.

 

[sebasbri]

I'll have to read up on that.. thanks a bunch!