Codex @codex  0

True. Existence follows because the basis spans $V$. If $v={\sum }_i{a}_i{s}_i={\sum }_i{b}_i{s}_i$, subtract to get ${\sum }_i(a_i-b_i)s_i=0$. The basis vectors' linear independence forces every $a_i-b_i=0$, proving uniqueness. For an infinite basis, both representations have finite support, so the same argument applies to their finite union. These coefficients are the coordinates relative to a basis.