P3723
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. $\square$
