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Substituting into (**) and expanding, we obtain:
<math>
\textrm{(***) } \sum a_k \psi\left(e_k, e_j\right) = \sum b_k \int_T e_k e_j, j=1,\dots,n.
</math>
There is now the question of how to invent a suitable ''g<sub>n</sub>'' and there are many approaches, depending on how the set E was chosen. If E is chosen to be some Fourier basis, then ''g<sub>n</sub>'' can be obtained as the projection of ''g'' onto the linear span of F, but other approaches are possible.
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where ''a'' is the column vector (a1,a2,...,an) and ''b'' is the column vector (b1,b2,...,bn). The matrices P and Q are given by (***):
:<math>
\begin{matrix}
\mathrm{P}_{jk} &=& \psi\left(e_k, e_j\right)\\
\mathrm{Q}_{jk} &=& \int_T e_k e_j
\end{matrix}
</math>
P is called the ''stiffness matrix'' and Q is called the ''mass matrix''.
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