Remark

If $u(x) = 0$ for all $x \in S$, then also $u(x) = 0$ for all $x \in M$. We shall sometimes refer to this result as the "principle of extension of linear identities" or "principle of extension by linearity".

In particular, to verify that a linear mapping $u : E \to F$ is of the form $v \circ \phi$, where $v : E / M \to F$ is linear and $\phi : E \to E / M$ is the canonical projection, it suffices to verify that $u(S) = 0$.