Affine solution space ($\{x:Ax=b\}=x_0+\ker A$)

If a linear system $Ax=b$ has one solution $x_0$, every other solution differs from it by an element of the null space: $Ax=b$ is equivalent to $A(x-x_0)=0$. Its solution set is therefore the displayed affine subspace, of dimension equal to the nullity. If the system is inconsistent, the solution set is empty, rather than a translate of the kernel.