Unique Representation by Ordered Basis

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Let $G$ be a unitary $R$-module.

Then $\sequence {a_n}$ is an ordered basis of $G$ if and only if:

For every $x \in G$ there exists one and only one sequence $\sequence {\lambda_n}$ of scalars such that $\ds x = \sum_{k \mathop = 1}^n \lambda_k a_k$.


Necessary Condition

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

Then every element of $G$ is a linear combination of $\set {a_1, \ldots, a_n}$, which is a generator of $G$, by Generated Submodule is Linear Combinations.

Thus there exists at least one such sequence of scalar.

Now suppose there were two such sequences of scalars: $\sequence {\lambda_n}$ and $\sequence {\mu_n}$.

That is, suppose $\ds \sum_{k \mathop = 1}^n \lambda_k a_k = \sum_{k \mathop = 1}^n \mu_k a_k$.


\(\ds \sum_{k \mathop = 1}^n \paren {\lambda_k - \mu_k} a_k\) \(=\) \(\ds \sum_{k \mathop = 1}^n \paren {\lambda_k a_k - \mu_k a_k}\)
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^n \lambda_k a_k - \sum_{k \mathop = 1}^n \mu_k a_k\)
\(\ds \) \(=\) \(\ds 0\)

So $\lambda_k = \mu_k$ for all $k \in \closedint 1 n$ as $\sequence {a_n}$ is a linearly independent sequence.


Sufficient Condition

Now suppose there is one and only one sequence $\sequence {\lambda_n}$ such that the condition holds.

It is clear that $\set {a_1, \ldots, a_n}$ generates $G$.

Suppose $\ds \sum_{k \mathop = 1}^n \lambda_k a_k = 0$.

Then, since also $\ds \sum_{k \mathop = 1}^n 0 a_k = 0$, we have, by hypothesis:

$\forall k \in \closedint 1 n: \lambda_k = 0$

Therefore $\sequence {a_n}$ is a linearly independent sequence.


Also see