Unique Linear Transformation Between Modules

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

Let $\sequence {a_n}$ be an ordered basis of $G$.

Let $\sequence {b_n}$ be a sequence of elements of $H$.

Then there is a unique linear transformation $\phi: G \to H$ satisfying:
 * $\forall k \in \closedint 1 n: \map \phi {a_k} = b_k$

Proof
By Isomorphism from R^n via n-Term Sequence, the mapping $\phi: G \to H$ defined as:
 * $\ds \map \phi {\sum_{k \mathop = 1}^n \lambda_k a_k} = \sum_{k \mathop = 1}^n \lambda_k b_k$

is well-defined.

Thus:
 * $\forall k \in \closedint 1 n: \map \phi {a_k} = b_k$

By Linear Transformation of Generated Module, $\phi$ is the only linear transformation whose value at $a_k$ is $b_k$ for all $k \in \closedint 1 n$.