Linear Combination of Sequence is Linear Combination of Set

Theorem
Let $G$ be an $R$-module.

Let $\sequence {a_k}_{1 \mathop \le k \mathop \le n}$ be a sequence of elements of $G$.

Let $b$ be an element of $G$.

Then:
 * $b$ is a linear combination of the sequence $\sequence {a_k}_{1 \mathop \le k \mathop \le n}$


 * $b$ is a linear combination of the set $\set {a_k: 1 \mathop \le k \mathop \le n}$
 * $b$ is a linear combination of the set $\set {a_k: 1 \mathop \le k \mathop \le n}$

Necessary Condition
By definition of linear combination of subset:


 * Every linear combination of $\sequence {a_k}_{1 \mathop \le k \mathop \le n}$ is a linear combination of $\set {a_k: 1 \mathop \le k \mathop \le n}$.

Sufficient Condition
Let $b$ be a linear combination of $\set {a_k: 1 \mathop \le k \mathop \le n} = \set {a_1, a_2, \ldots, a_n}$.

Then there exists:
 * a sequence $\sequence {c_j}_{1 \mathop \le j \mathop \le m}$ of elements of $\set {a_1, a_2, \ldots, a_n}$

and:
 * a sequence $\sequence {\mu_j}_{1 \mathop \le j \mathop \le m}$ of scalars such that:


 * $\displaystyle b = \sum_{j \mathop = 1}^m \mu_j c_j$

For each $k \in \closedint 1 n$, let $\lambda_k$ be defined as follows.

If:
 * $a_k \in \set {c_1, c_2, \ldots, c_m}$

and:
 * $a_i \ne a_j$ for all indices $i$ such that $1 \le i < k$

let $\lambda_k$ be the sum of all scalars $\mu_j$ such that $c_j = a_k$.

If:
 * $a_k \notin \set {c_1, c_2, \ldots, c_m}$

or:
 * $a_i = a_j$ for some index $i$ such that $1 \le i < k$

let $\lambda_k = 0$.

It follows that:
 * $\displaystyle b = \sum_{j \mathop = 1}^m \mu_j c_j = \sum_{k \mathop = 1}^n \lambda_k a_k$

Let $\sequence {a_k}_{1 \mathop \le k \mathop \le n}$ and $\sequence {b_j}_{1 \mathop \le j \mathop \le m}$ be sequences of elements of $G$ such that $\set {a_1, a_2, \ldots, a_n}$ and $\set {b_1, b_2, \ldots, b_m}$ are identical.

Then as a consequence of the above:
 * an element is a linear combination of $\sequence {a_k}_{1 \mathop \le k \mathop \le n}$


 * it is a linear combination of $\set {a_k: 1 \mathop \le k \mathop \le n}$
 * it is a linear combination of $\set {a_k: 1 \mathop \le k \mathop \le n}$