Unique Linear Transformation Between Modules

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Theorem

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:

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


Corollary

Let $G$ be a finite-dimensional $K$-vector space.

Let $H$ be a $K$-vector space (not necessarily finite-dimensional).

Let $\sequence {a_n}$ be a linearly independent sequence of vectors of $G$.

Let $\sequence {b_n}$ be a sequence of vectors 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:

$(1): \quad \ds \map \phi {\sum_{j \mathop = 1}^n \lambda_j \times_G a_j} = \sum_{j \mathop = 1}^n \lambda_j \times_H b_j$

is well-defined.

\(\ds \forall k \in \closedint 1 n: \, \) \(\ds \map \phi {\sum_{j \mathop = 1}^n \lambda_{j k} \times_G a_j}\) \(=\) \(\ds \sum_{j \mathop = 1}^n \lambda_{j k} \times_H b_j\) from $(1)$
\(\ds \leadsto \ \ \) \(\ds \map \phi {\sum_{j \mathop = 1}^n \delta_{j k} \times_G a_j}\) \(=\) \(\ds \sum_{j \mathop = 1}^n \delta_{j k} \times_H b_j\) setting $\lambda_{j k} = \delta_{j k}$, where $\delta_{j k}$ is Kronecker Delta
\(\ds \leadsto \ \ \) \(\ds \map \phi {a_k}\) \(=\) \(\ds b_k\) Definition of Kronecker Delta


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:

\(\ds \map \phi {x +_G y}\) \(=\) \(\ds \map \phi {\sum_{k \mathop = 1}^n \lambda_k \times_G a_k +_G \sum_{k \mathop = 1}^n \mu_k \times_G a_k}\) Definition of $x$ and $y$
\(\ds \) \(=\) \(\ds \map \phi {\sum_{k \mathop = 1}^n \paren {\lambda_k + \mu_k} \times_G a_k}\) Module Axiom $\text M 2$: Distributivity over Scalar Addition
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^n \paren {\lambda_k + \mu_k} \times_H b_k\) from $(1)$
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^n \lambda_k \times_H b_k +_H \sum_{k \mathop = 1}^n \mu_k \times_H b_k\) Module Axiom $\text M 2$: Distributivity over Scalar Addition
\(\ds \) \(=\) \(\ds \map \phi x +_H \sum_{k \mathop = 1}^n \map \phi y\) Definition of $x$ and $y$


and:

\(\ds \map \phi {\lambda \times_G x}\) \(=\) \(\ds \map \phi {\lambda \times_G \sum_{k \mathop = 1}^n \lambda_k \times_G a_k}\) Definition of $x$
\(\ds \) \(=\) \(\ds \map \phi {\sum_{k \mathop = 1}^n \lambda \times_G \paren {\lambda_k \times_G a_k} }\) Module Axiom $\text M 1$: Distributivity over Module Addition
\(\ds \) \(=\) \(\ds \map \phi {\sum_{k \mathop = 1}^n \paren {\lambda \lambda_k} \times_G a_k}\) Module Axiom $\text M 3$: Associativity
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^n \paren {\lambda \lambda_k} \times_H b_k\) from $(1)$
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^n \lambda \times_H \paren {\lambda_k \times_H b_k}\) Module Axiom $\text M 1$: Distributivity over Module Addition
\(\ds \) \(=\) \(\ds \lambda \times_H \map \phi x\) Definition of $x$


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

$\blacksquare$


Sources

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