Consider a linear regression model $Y = \beta_0 + \beta_1 x + \epsilon$. We learned about generating a residual vs predictor plot to examine the performance of the regression model. If there is a pattern instead of a random scatter plot, then we might want to change the model by adding new predictors or changing to a nonlinear model.
Does this mean if the true model is indeed $Y = \beta_0 + \beta_1 x$, i.e. I correctly chose the predictors to study and correctly decide to study linear relationship, then the residual $Y - \hat Y$ is independent from $X$.
I know that $Y - \hat Y = Y - PY$, where $ P = X(X'X)^{-1}X'$. But I don't know how to show $Y - \hat Y$ is independent from $X$.
Also I'm considering the general case $\epsilon$ is any symmetric distribution, does not have to be Gaussian.