= Coordinate vector
{title2=$[v]_{\mathcal B}$}
{wiki}
For an <ordered basis> $\mathcal B=(b_1,\ldots,b_n)$, the coordinate vector $[v]_{\mathcal B}=(a_1,\ldots,a_n)^T$ records the unique <coefficients> satisfying $v=\sum_i a_i b_i$. <Change of basis> changes this representation while preserving the underlying <vector>.
For $b_1=(1,0)^T$, $b_2=(1,1)^T$, the <vector> $v=(3,1)^T$ has <coordinate vector> $(2,1)^T$, since $2b_1+b_2=v$. These are the same coefficients extracted by the <dual basis> in its worked example. The <vector> is independent of the chosen <basis>; its list of <coordinates> is not. In an <orthonormal eigenbasis>, the <coordinates> additionally separate the action of a <self-adjoint operator> into independent <scalar multiplications>.
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