# 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 $S = \set {x \in G: \map \phi x = \map \psi x}$ is a submodule of $G$.

## Proof

Let $x, y \in S$.

Let $\lambda \in R$.

Then:

 $\ds \map \phi {x + y}$ $=$ $\ds \map \phi x + \map \phi y$ Definition of Linear Transformation $\ds$ $=$ $\ds \map \psi x + \map \psi y$ $x, y \in S$ $\ds$ $=$ $\ds \map \psi {x + y}$ Definition of Linear Transformation $\ds \map \phi {\lambda \circ x}$ $=$ $\ds \lambda \circ \map \phi x$ Definition of Linear Transformation $\ds$ $=$ $\ds \lambda \circ \map \psi x$ $x \in S$ $\ds$ $=$ $\ds \map \psi {\lambda \circ x}$ Definition of Linear Transformation

Hence $x + y, \lambda \circ x \in S$.

By Submodule Test, $S$ is a submodule of $G$.

$\blacksquare$