Theorem 8

Let

be a system of scalar linear equations on a vector space E over a field K. For this system to have at least one solution, it is necessary that, for every family $(\rho_i)$ of scalars of finite support such that $\sum_i x_i^* \rho_i = 0,\ \sum_i \eta_i \rho_i = 0.$ If the rank of the system (21) is finite, this condition is also sufficient.

The condition is obviously necessary. It says that, if $F'$ is the subspace of $E^*$ generated by the family $(x_i^*)$, there exists a linear mapping $f : F' \to K_d$ such that $f(x_i^*) = \eta_i$ for all $i \in I$. If $F'$ is of finite dimension $r$, $F'$ is the orthogonal of a subspace $F$ of $E$ of codimension $r$ (no. 5, Theorem 7) and $F'$ is identified with the dual of $E/F$ (§ 2, no. 6, Corollary to Proposition 9); $f$ is therefore an element of the bidual $(E/F)**$. As $E/F$ is finite-dimensional there exists one and only one element $y \in E/F$ such that $f(x^*) = \langle y, x^* \rangle$ for all $x^* \in F'$ (no. 5, Theorem 6). The solutions of (21) are then the $x \in E$ whose canonical image in $E/F$ is $y$.