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:

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

Proof

Existence

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

$\Box$

Uniqueness

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

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

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

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

Thus the theorem follows from Expression of Vector as Linear Combination from Basis is Unique.

$\blacksquare$