Unique Linear Transformation Between Modules

Theorem
Let $\struct {G, +_G, \times_G}_R$ and $\struct {H, +_H, \times_H}_R$ 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 Unique Representation by Ordered Basis, 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:
 * $\forall k \in \closedint 1 n: \map \phi {a_k} = b_k$

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:


 * $\ds x = \sum_{k \mathop = 1}^n \lambda_k a_k$


 * $\ds y = \sum_{k \mathop = 1}^n \mu_k a_k$

where:
 * $a_k$ are elements of $\sequence {a_n}$
 * $\lambda_k$ and $\mu_k$ are elements of $R$.

So:

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