Prove the part of the theorem which lets w be any solution of Ax = b, and defines vh = 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.

It is proved that is a solution of Ax=b.

Step-by-step explanation:

As given let w and p be a solution of Ax=b. Then,

and

Defining,

To show is a solution of Ax=b, we have to show . Then,

       (By putting value of )

      (By distributive law)

                    (Since w and p are solution of Ax=b)

Hence become a solution of the homogenous system Ax=b.

Since w and p are solution of Ax=b, and any linear combination of the solutions again a solution of Ax=b, this shows that every solution of Ax=b is of the form  where is a solution of Ax=0.

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