Expression of Vector as Linear Combination from Basis is Unique/General Result

Theorem
Let $V$ be a vector space over a division ring $R$.

Let $B$ be a basis for $V$.

Let $x \in V$.

Then there is a unique finite subset $C$ of $R \times B$ such that:


 * $\displaystyle x = \sum_{(r, v) \in C} r \cdot v$ and
 * $\forall (r,v) \in C: r \ne 0_R$

Existence
The existence of $C$ follows immediately from the definition of a basis.

Uniqueness
Let $C, D$ be finite subsets of $R \times B$ satisfying the requirements.

Let $Q = \{ v \mid \exists r \in R: (r,v) \in C \cup D \}$.

Let $V' = \operatorname{span}(Q)$.

Then $V'$ is a finite-dimensional vector space with basis $Q$ and $x \in V'$.

Thus the theorem follows from the finite-dimensional case.