a(f, h) = b(h), \qquad (h \in V),$$ where $$a(u, v) := \int_{\Omega} \nabla u(x) \cdot \nabla v(x) {\, \mathrm{d}}x, \qquad b(u) := \int_{\Omega} g(x) \, u(x) {\, \mathrm{d}}x.$$

If we now choose functions $h_1, \ldots, h_m \in V$, we can look at the approximate solution $\phi$ to in the subspace $V_m := \langle h_1, \ldots, h_m \rangle_{{\mathbf{R}}} \subseteq V$ spanned by $h_1, \ldots, h_m$. Because $\phi \in V_m$, there exist coefficients $\phi_1, \ldots, \phi_m \in {\mathbf{R}}$ such that $\phi = \sum_{i = 1}^m \phi_i \, h_i$, and therefore, $$b(h_j) = a(\phi, h_j) = a \left( \sum_{i = 1}^m \phi_i \, h_i, h_j \right)
 = \sum_{i = 1}^m \phi_i \, a(h_i, h_j), \qquad (1 \leq j \leq m).$$ So solving in $V_m$ for $\phi$ amounts to solving the linear system $$\sum_{i = 1}^m a(h_i, h_j) \, \phi_i = b(h_j), \qquad (1 \leq j \leq m),$$ for $(\phi_1, \ldots, \phi_m) \in {\mathbf{R}}^m$. By choosing $h_1, \ldots, h_m$ such that their supports only have a small overlap we create a sparse matrix $A \in {\mathbf{R}}^{m \times m}$ with entries $a_{i \, j} := a(h_j, h_i)$ that are nonzero only for the $1 \leq i, j \leq m$ where $\supp h_i \cap \supp 