Isomorphism from R^n via n-Term Sequence

Theorem
Let $G$ be a unitary $R$-module.

Let $\left \langle {a_n} \right \rangle$ be an ordered basis of $G$.

Let $R^n$ be the $R$-module $R^n$.

Let $\psi: R^n \to G$ be defined as:
 * $\displaystyle \psi \left({\left \langle {\lambda_n} \right \rangle}\right) = \sum_{k=1}^n \lambda_k a_k$

Then $\psi$ is an isomorphism.

Proof

 * By Unique Representation by Ordered Basis, $\psi$ is a bijection.


 * We have:

and we have:

thus proving that $\phi$ is also a homomorphism.