Solution
= Solution
Full column rank gives the least-squares estimator
$$
\boxed{\widehat\theta=(X^TX)^{-1}X^TY.}
$$
Write $A=(X^TX)^{-1}X^T$. Since $AX=I$ and $\operatorname{Cov}(\epsilon)=\sigma^2I$, we have $\widehat\theta=\theta+A\epsilon$ and
$$
\boxed{\operatorname{Cov}(\widehat\theta)=\sigma^2AA^T
=\sigma^2(X^TX)^{-1}.}
$$
This calculation needs the stated uncorrelated equal-<variance> errors, but not normality.