Given an ordered basis $B=(b_1,\ldots,b_n)$, the coordinate vector $[v]_B=(c_1,\ldots,c_n)^T$ is defined by $v=\sum_jc_jb_j$. Basis independence makes the coefficients unique. For $B=((1,0),(1,1))$, $[(5,3)]_B=(2,3)^T$.
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