Editing System of linear equations (section)

===Relation to nonhomogeneous systems===
There is a close relationship between the solutions to a linear system and the solutions to the corresponding homogeneous system:
: <math>A\mathbf{x}=\mathbf{b}\qquad \text{and}\qquad A\mathbf{x}=\mathbf{0}.</math>
Specifically, if '''p''' is any specific solution to the linear system {{nowrap|1=''A'''''x''' = '''b'''}}, then the entire solution set can be described as
: <math>\left\{ \mathbf{p}+\mathbf{v} : \mathbf{v}\text{ is any solution to }A\mathbf{x}=\mathbf{0} \right\}.</math>
Geometrically, this says that the solution set for {{nowrap|1=''A'''''x''' = '''b'''}} is a [[translation (geometry)|translation]] of the solution set for {{nowrap|1=''A'''''x''' = '''0'''}}. Specifically, the [[flat (geometry)|flat]] for the first system can be obtained by translating the [[Euclidean subspace|linear subspace]] for the homogeneous system by the vector '''p'''.

This reasoning only applies if the system {{nowrap|1=''A'''''x''' = '''b'''}} has at least one solution. This occurs if and only if the vector '''b''' lies in the [[image (mathematics)|image]] of the [[linear transformation]] ''A''.