Theorem: Unique Linear Transformation Between Modules

Let $\struct {G, +_G, \times_G}_R$ and $\struct {H, +_H, \times_H}_R$ be unitary $R$-modules.
Let $\sequence {a_k}_{1 \mathop \le k \mathop \le n}$ be an ordered basis of $G$.
Let $\sequence {b_k}_{1 \mathop \le k \mathop \le n}$ be a sequence of $n$ elements of $H$.
Then there exists a unique linear transformation $\phi: G \to H$ satisfying:
:$\forall k \in \closedint 1 n: \map \phi {a_k} = b_k$

Proof:
By Unique Representation by Ordered Basis, the mapping $\phi: G \to H$ defined as:
:$(1): \quad \ds \map \phi {\sum_{j \mathop = 1}^n \lambda_j \times_G a_j} = \sum_{j \mathop = 1}^n \lambda_j \times_H b_j$
is well-defined.
{{begin-eqn}}
{{eqn | q = \forall k \in \closedint 1 n
      | l = \map \phi {\sum_{j \mathop = 1}^n \lambda_{j k} \times_G a_j}
      | r = \sum_{j \mathop = 1}^n \lambda_{j k} \times_H b_j
      | c = from $(1)$
}}
{{eqn | ll= \leadsto
      | l = \map \phi {\sum_{j \mathop = 1}^n \delta_{j k} \times_G a_j}
      | r = \sum_{j \mathop = 1}^n \delta_{j k} \times_H b_j
      | c = setting $\lambda_{j k} = \delta_{j k}$, where $\delta_{j k}$ is Kronecker Delta
}}
{{eqn | ll= \leadsto
      | l = \map \phi {a_k}
      | r = b_k
      | c = {{Defof|Kronecker Delta}}
}}
{{end-eqn}}
To verify that $\phi$ is a linear transformation, we need to show the following:
:$(1): \quad \forall x, y \in G: \map \phi {x +_G y} = \map \phi x +_H \map \phi y$
:$(2): \quad \forall x \in G: \forall \lambda \in R: \map \phi {\lambda \times_G x} = \lambda \times_H \map \phi x$
Let $x, y \in G$ be arbitrary, such that:
:$\ds x = \sum_{k \mathop = 1}^n \lambda_k \times_G a_k$
:$\ds y = \sum_{k \mathop = 1}^n \mu_k \times_G a_k$
where:
:$a_k$ are elements of $\sequence {a_n}$
:$\lambda_k$ and $\mu_k$ are elements of $R$.
So:
{{begin-eqn}}
{{eqn | l = \map \phi {x +_G y}
      | r = \map \phi {\sum_{k \mathop = 1}^n \lambda_k \times_G a_k +_G \sum_{k \mathop = 1}^n \mu_k \times_G a_k}
      | c = Definition of $x$ and $y$
}}
{{eqn | r = \map \phi {\sum_{k \mathop = 1}^n \paren {\lambda_k + \mu_k} \times_G a_k}
      | c = {{Module-axiom|2}}
}}
{{eqn | r = \sum_{k \mathop = 1}^n \paren {\lambda_k + \mu_k} \times_H b_k
      | c = from $(1)$
}}
{{eqn | r = \sum_{k \mathop = 1}^n \lambda_k \times_H b_k +_H \sum_{k \mathop = 1}^n \mu_k \times_H b_k
      | c = {{Module-axiom|2}}
}}
{{eqn | r = \map \phi x +_H \sum_{k \mathop = 1}^n \map \phi y
      | c = Definition of $x$ and $y$
}}
{{end-eqn}}
and:
{{begin-eqn}}
{{eqn | l = \map \phi {\lambda \times_G x}
      | r = \map \phi {\lambda \times_G \sum_{k \mathop = 1}^n \lambda_k \times_G a_k}
      | c = Definition of $x$
}}
{{eqn | r = \map \phi {\sum_{k \mathop = 1}^n \lambda \times_G \paren {\lambda_k \times_G a_k} }
      | c = {{Module-axiom|1}}
}}
{{eqn | r = \map \phi {\sum_{k \mathop = 1}^n \paren {\lambda \lambda_k} \times_G a_k}
      | c = {{Module-axiom|3}}
}}
{{eqn | r = \sum_{k \mathop = 1}^n \paren {\lambda \lambda_k} \times_H b_k
      | c = from $(1)$
}}
{{eqn | r = \sum_{k \mathop = 1}^n \lambda \times_H \paren {\lambda_k \times_H b_k}
      | c = {{Module-axiom|1}}
}}
{{eqn | r = \lambda \times_H \map \phi x
      | c = Definition of $x$
}}
{{end-eqn}}
By Linear Transformation of Generated Module, $\phi$ is the only linear transformation whose value at $a_k$ is $b_k$ for all $k \in \closedint 1 n$.
{{Qed}}