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 $$\psi \left({\left \langle {\lambda_n} \right \rangle}\right) = \sum_{k=1}^n \lambda_k a_k$$

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.