For question 4.a) is it enough to say that because the model is non-singular, it is invertible, so therefore ?Xj??? is a projection matrix so then the formula for the least square estimator beta is satisfied?

I haven't taken much linear algebra and I've never taken a proofs based course before so I apologize if my reasoning is completely incorrect. Is it actually that ?Xj??? is only a residual and not a projection matrix?

3. Let Y = X1B1 + X262 + & be a linear model with nonsingular design. Find the condition under which the least square estimator Bi for Bi is the same as one from regressing Y only on X1. 4. Let Y = D._, BjX; + e be a linear model with nonsingular design, and B = (B1, . . ., Bk )T the least square estimator. For each j, let V; denote the subspace spanned X; onto Vj. by all covariates excluding X;, and X" = X; - Pv,X; be the residual of projecting (a). Show that (Y, X-) X-112 (b). Find the variance of B; and the covariance between B; and B; for each pair i f j. 1, ..., k}. (c). Use the result in (b) to express the matrix (XTX)-1 in terms of {X- : j = 5. Let V = span(X1, . .., Xk) have dimension k. Under the same notation of the previous problem, show that {X; : j = 1, ..., k} also forms a basis for V. For each Y ER", find the coefficients a; in the expansion PVY = >ajx