Elements of Module with Equal Images under Linear Transformations form Submodule

Theorem
Let $$G$$ and $$H$$ be $R$-modules.

Let $$\phi$$ and $$\psi$$ be linear transformations from $$G$$ into $$H$$.

Then the set $$S = \left\{{x \in G: \phi \left({x}\right) = \psi \left({x}\right)}\right\}$$ is a submodule of $$G$$.

Also see
Compare with Homomorphisms on the Same Groups.