# Find $V(\tilde\beta|X)$

In the linear regression $$Y=X\beta+\epsilon$$, with $$E(\epsilon_i|x_i)=0$$, it is known that the true $$\beta$$ satisfies the restriction $$M\beta=0$$, where $$M$$ is a $$q \times k$$ matrix with $$q. $$\hat\beta$$ is estimator for $$\beta$$, where $$\hat\beta=(X^TX)^{-1}X^TY$$.

Consider the estimator: $$\tilde\beta=\hat\beta-(X^TX)^{-1}M^T[M(X^TX)^{-1}M^T]^{-1}M\hat\beta$$. How to find $$V(\tilde\beta|X)$$? Also, what is an expression for a valid standard error for the elements of $$\tilde\beta$$?

I think I have to first write $$\tilde\beta$$ as a linear function of $$\hat\beta$$, but I am stuck on how to proceed. I have also shown that $$M\tilde\beta=0$$.