Prove the part of the theorem which lets w be any solution o

Prove the part of the theorem which lets w be any solution of Ax = b, and defines v_h = w - p. Show that v_h is a solution of Ax = 0. This shows that every solution of Ax = b has the form w = p + v_h, with p a particular solution of Ax = b and v_h a solution of Ax = 0. Let w and p be solutions of Ax = b. Substitute for v_h from the equation w = p + v_h. Av_h = A

Solution

Let w be a solution of the equation Ax = b and let p be a particular solution of Ax = b. Also, let v_h be a solution of Ax = 0. Then Ap = b and Av_h = 0. Now, if Ap = b and Av_h = 0 then A(p + v_h) = Ap + Av_h = b + 0 = b. On the other hand, if Aq = b is another solution of Ax = b, then A(p q) = 0 and p q = v_h is a solution of Ax = 0. So q = p + v_h. Thus any soltion of the equation Ax = b has the form w = p + v_h where p is a particular solution of Ax = b and v_h a solution of Ax = 0.