Isomorphism from R^n via n-Term Sequence

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

Let $\sequence {a_k}_{1 \mathop \le k \mathop \le n}$ be an ordered basis of $G$.

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

Let $\psi: R^n \to G$ be defined as:
 * $\displaystyle \map \psi {\sequence {\lambda_k}_{1 \mathop \le k \mathop \le n} } = \sum_{k \mathop = 1}^n \lambda_k a_k$

Then $\psi$ is an isomorphism.

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

We have:

and we have:

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