Let Ax b be any consistent system of linear equations and le

Let Ax b be any consistent system of linear equations, and let X_1 be a fixed solution. Show that every solution to the system can be written in the form x = x_1 + x_0, where x_0 is a solution to Ax = 0. Show also that every matrix of this form is a solution.

Solution

We are given x1 is a solution to Ax = b

So, Ax1 =b ----(1)

We are given x0 is a solution to Ax = 0

Ax0 = 0 ----(2)

Adding 1 and 2 : Ax1 +Axo = b +0

A( x1 +x0) = b

So, solution to the above equation is x = x1 +x0