Linear_Transformation_from_Ordered_Basis_less_Kernel

Let $G$ and $H$ be unitary $R$-modules. Let $\phi: G \to H$ be a non-zero linear transformation. Let $G$ be $n$-dimensional. Let $\left \langle {a_n} \right \rangle$ be any ordered basis of $G$ such that $\left\{{a_k: r + 1 \le k \le n}\right\}$ is the basis of the kernel of $\phi$. Then $\left \langle {\phi \left({a_r}\right)} \right \rangle$ is an ordered basis of the image of $\phi$.