The basis of the real space \(\mathbb{R}^n\) is the set

\begin{equation*} E = \{\vec{e}_1, \ldots, \vec{e}_n\}, \end{equation*}

where \(\vec{e}_i\) is the \(i\)-th column of the identity matrix \(I_n\), so

\begin{equation*} \vec{e}_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \\ \vdots \\ 0 \end{pmatrix}, \vec{e}_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \\ \vdots \\ 0 \end{pmatrix}, \vec{e}_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \\ \vdots \\ 0 \end{pmatrix}, \ldots, \vec{e}_n = \begin{pmatrix} 0 \\ 0 \\ 0 \\ \vdots \\ 1 \end{pmatrix}. \end{equation*}