Characterizing Basis by Unique Representation Theorem

$S$ is a basis of $V$ $⟺$ $\forall \overrightarrow{v}\in V,$ $\overrightarrow{v}$ has a Unique Representation in $span(S)$.

Proof

This proof relies heavily on linear independent sets

Proving $⟹$

1. Assume $S$ is a basis of $V$
2. Pick $\overrightarrow{v}\in V$, we know $span(S)=V$
3. Then, $\overrightarrow{v}=a_1{\overrightarrow{v}}_1+a_2{\overrightarrow{v}}_2+\dots a_n{\overrightarrow{v}}_n\in span(S)$ Where $a_i\in F$, $v_i\in S$
4. The vectors in $S$ are lin indep, and so this representation is unique

Proving $⟸$

1. Assume every vector in $V$ has a unique representation in $span(S)$
2. Then, $V\subset span(S)$
3. Therefore, $V=span(S)$. It follows that $S$ is a spanning set of $V$
4. We get that S is Linearly Independent because this representation is unique Therefore, $S$ is a basis.