Solution
= Solution
\b[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>.