Unique Linear Transformation Between Modules

Theorem
Let $$G$$ and $$H$$ be unitary $R$-modules.

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

Let $$\left \langle {b_n} \right \rangle$$ be a sequence of elements of $$H$$.

Then there is one and only one linear transformation $$\phi: G \to H$$ satisfying $$\forall k \in \left[{1 \,. \, . \, n}\right]: \phi \left({a_k}\right) = b_k$$

Poof
By Isomorphism from R^n via n-Term Sequence, the mapping $$\phi: G \to H$$ defined as $$\phi \left({\sum_{k=1}^n \lambda_k a_k}\right) = \sum_{k=1}^n \lambda_k b_k$$ is well-defined.

Thus $$\forall k \in \left[{1 \,. \, . \, n}\right]: \phi \left({a_k}\right) = b_k$$.

By Linear Transformation of Generated Module, $$\phi$$ is the only linear transformation whose value at $$a_k$$ is $$b_k$$ for all $$k \in \left[{1 \,. \, . \, n}\right]$$.