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Intuition: Standard basis
The vector notation
\begin{equation*} \begin{pmatrix} v_1 \\ \vdots \\ v_n \end{pmatrix} \end{equation*}is shorthand for a linear combination of the standard basis
\begin{equation*} v_1 \vec{e}_1 + v_2 \vec{e_2}. \end{equation*}