quality:corruption
Relation to nonhomogeneous systems

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

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

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

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