You may also Find vector b orthogonally
$$\mathbf{e’e} = \mathbf{y’y – 2b’_0X’y + b’_0X’Xb_0}$$
or,
$$\mathbf{S(b)} = \mathbf{y’y – 2b’_0X’y + b’_0X’Xb_0}$$
vector $b$ is a minimum when
$$\dfrac{\partial S(b) }{\partial b} = \mathbf{- 2X’y + 2X’Xb} = 0 $$
$$\therefore \mathbf{X’Xb = X’y}$$
$$ \boxed{\large\widehat\beta = (X’X)^{-1} X’ y}$$
$$ \boxed{\large\widehat \beta = \beta + (X’X)^{-1} X’ u} $$
$$\dfrac{\partial S(b) }{\partial b} = \mathbf{- 2X’y + 2X’Xb} = 0$$
$$\therefore \mathbf{X’Xb} = \mathbf{X’y}$$
$$\large\widehat\beta = (X’X)^{-1} X’ y$$
$$\boxed{\large\widehat\beta = \beta + (X’X)^{-1} X’ u}$$