Proposition 14

If $x_0$ is a solution of the linear equation $u(x) = y_0$, the set of solutions of this equation is equal to the set of elements $x_0 + z$, where $z$ runs through the set of solutions of the associated homogeneous equation $u(x) = 0$.

The relation $u(x) = y_0$ may be written as $u(x) = u(x_0)$, which is equivalent to $u(x - x_0) = 0$.

In other words, if the equation $u(x) = y_0$ has at least one solution $x_0$, the set of its solutions is the set $x_0 + \overline{u}(0)$, obtained by translation from the kernel $\overline{u}(0)$ of $u$. Observe that $\overline{u}(0)$, being a submodule, is never empty, since it contains 0 (called the zero solution, or trivial solution, of the homogeneous equation $u(x) = 0$).

By virtue of Proposition 14, for a linear equation $u(x) = y_0$ to have exactly one solution, it is necessary and sufficient that it have at least one solution and that $\overline{u}(0) = \{0\}$ (in other words, that the associated homogeneous equation have no non-zero solution, or also that $u$ be injective); in this case, for all $y \in F$, the equation $u(x) = y$ has at most one solution.