One can replace y with in = (X'X) X'y and obtain the same OLS fitted coefficients. Show this without using the algebraic solution for 8 by demonstrating the more general result that

BB argminy - X||2 = argmin | X81 } In your demonstration, justify each of the following equalities: min y X8 min ||-|| = Col(X) (2.19) = min *Col(X) - (2.20) = ||- |+min ||-|| (2.21) = Col(X) y-min-X81 Does the argument rest on whether X is full-column rank? (2.22)