In OLS with fixed design, we project $Y$ onto the column space of design matrix $X$, which is $\mathcal{C}(X)$. The residual $Y-PY = (I-P)Y$ is orthogonal $\mathcal{C}(X)$.
Further more, $(I-P)$ is a projection onto $\mathcal{C}(X)^{\perp}$, so $(I-P)X = 0$.
Now consider a random design matrix $X$, where each row is i.i.d a $p$-dimensional sub-Gaussian random vector. I understand the space $\mathcal{C}(X)$ is random. May I ask if the classic property about projection matrix $(I-P)$ still holds in this context?