Consider a data generating mechanism of $Y_i = \mu(X_i)+ \epsilon_i$, $\epsilon_i \sim N(0,1)$. $\mu(X)$ is a nonlinear function. Suppose $X$ is also a random variable with $E[X]=0$ and $Var[X] = \sigma_X$.
We fit a model $Y_i = X_i \beta$ using the Least squares method. Let $$\beta^{*} = argmin_{\beta} E(\mu(X_i) - X_i \beta)^2.$$
What is $Cov(\mu(X), X\beta^{*})$? Is it true that $Cov(\mu(X), X\beta^{*})=Var(X\beta^{*})$?
The context of this question is that I came across this expression when I was reading the following lecture note (pp.7, the note on the bottom).
https://web.stanford.edu/~swager/stats361.pdf
It says that $X\beta^{*}$ is the projection of $\mu(X)$ onto the linear span of the features $X$ but I could not figure out what this means.
Any help would be highly appreciated!