Uniqueness of Expression of a Vector as a Linear Combination

Uniqueness of Expression of a Vector as a Linear Combination

This can be summarized with:
($Sx = 0$ has only the trivial solution)
$\Leftrightarrow$
($v \in \text{span}(S) \Rightarrow Sx = v$ has an unique solution)
Hint: Break into two parts. Reductio ad absurdum helps.