let Ax=b be any consistent system of linear equations, and let x1 be a fixed solution. show that every solution to the system can be written in the form x=x1+x0, where x0 is a solution to Ax=0. show that every matrix of this form is a solution.

There is a close relationship between the solutions to a linear system and the solutions to the corresponding homogeneous system:

Ax=b and Ax+0

Specifically, if x1 is any specific solution to the linear system Ax = b, then the entire solution set can be described as

x1 + x0 : x0 is any solution to Ax=0

Geometrically, this says that the solution set for Ax = b is a translation of the solution set for Ax = 0. Specifically, the flat for the first system can be obtained by translating the linear subspace for the homogeneous system by the vector x1.

This reasoning only applies if the system Ax = b has at least one solution. This occurs if and only if the vector b lies in the image of the linear transformation A.