Is there an error in the reasoning in these lecture notes from a linear models course at EPFL? The author aims to establish that under the standard fixed design linear regression model with gaussian errors, the coefficient estimate $\hat{\beta}$ is independent of the residual variance estimate $S^2$. He writes,
"If $e=y-X\hat{\beta}$ is independent of $\hat{y}=X\hat{\beta}$, then $S^2=e^Te/(n-p)$ will obviously be independent of $\hat{\beta}$. Now notice that: ..."
And he goes on to show that $e$ is independent of $\hat{y}$ in the usual way, one is a projection onto the column space of $X$, the other is the projection onto its orthogonal complement, and the distributions are jointly gaussian so uncorrelatedness implies independence.
But I don't follow what he suggests is "obvious", i.e., why $e$ being independent of $X\hat{\beta}$ implies $e$ is independent of $\hat{\beta}$, since $X$ need not be invertible. In fact, while it is easy to visualize the independence of $e$ and $\hat{y}$ in terms of projections, I don't know how to visualize the relationship between $e$ and $\hat{\beta}$, the projection scaled by the squared norm of $X$, in a way that suggests uncorrelatedness.
