Let $R$ be an integral domain, let $R^n$ be a free $R$-module with basis $\{v_i\}_i^n$, let $\phi:R^n \to R^n: v_j \mapsto \sum_i^n m_{ij}v_i$ be an $R$-module homomorphism, let $F$ be $R$-module isomorphic to $R^n$, and let $\{\sum^n_i m_{ij}x_i\}_j^n$ be a basis for $F$.
Question: How would you argue $\phi$ is an isomorphism (which would establish $\{\sum_i^n m_{ij}v_i\}^n_j$ is a basis)? Clearly, there's an evident relationship between $\{\sum_i^n m_{ij}v_i\}_j$ and $\{\sum_i^n m_{ij}x_i\}_j$, if we could somehow use the fact that the coefficients are shared with elements we already know to be linearly independent and span the entire space.