Let $I_N \in \mathbb{R}^{N \times N}$ be a D5621: Real identity matrix such that

(i)	$e_1, \ldots, e_N \in \mathbb{R}^{N \times 1}$ are each a D5200: Real column matrix
(ii)	\begin{equation} I_N = \begin{bmatrix} e_1 & e_2 & \cdots & e_N \end{bmatrix} \end{equation}

Let $x \in \mathbb{R}^{N \times 1}$ be a real column matrix such that \begin{equation} x = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_N \end{bmatrix} \end{equation} Then \begin{equation} x = \sum_{n = 1}^N x_n e_n \end{equation} Since $x \in \mathbb{R}^{N \times 1}$ was arbitrary, the proof is complete. $\square$