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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

\[\begin{equation*}
  v_1 \vec{e}_1 + v_2 \vec{e_2}.
\end{equation*}
\]

— https://thenumb.at/Functions-are-Vectors/