= Uniqueness of linear combination coefficients
{title2=$\sum_i a_i v_i=\sum_i b_i v_i\Longrightarrow a_i=b_i$}
A family is <linearly independent> if and only if every <vector> in its <linear span> has unique <coefficients> as a <linear combination> of that family. For a short proof, subtract two representations to get $\sum_i(a_i-b_i)v_i=0$; <linear independence> forces each difference to vanish. Conversely, uniqueness applied to the <zero vector> forces every <linear relation> to use only zero <coefficients>. This explains the practical role of <linear independence>: a <coordinate vector> relative to an <ordered basis> is unambiguous.
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