I am trying to show that $$\sum_{i=1}^ne_i = 0$$
I have two hints, so to speak:
$$ HX = X$$ where $H$ is the hat matrix, and that $$\sum_{i=1}^ne_i = e'1$$
My solution is as follows:
$$e'1 = Y'(I-H)1=[(X\beta)' - (X\beta)'H]1=(\beta'X' - \beta'X'X(X'X)^{-1}X')1 $$ $$=(\beta'X' - \beta'X')1 = 01 = 0 $$
Seems simple enough but this implies that $e'$ and by extension $e$ is always a vector of zeros, which seems counter-intuitive.
If the sum of the residuals is zero , it does not necessarily mean that $\mathbf e'$ is a vector of zeros.
Example:
$\mathbf e'\cdot \mathbf 1=\begin{pmatrix}-2&-3&5\end{pmatrix}\cdot \begin{pmatrix}1\\ 1\\ 1\end{pmatrix}=-2-3+5=0$