Solution set of a linear system ($\{x:Ax=b\}=x_0+\ker A$)

For a consistent linear system $Ax=b$, choose one solution $x_0$. By linearity, $Ax=b$ if and only if $A(x-x_0)=0$, so its solution set is the translate $x_0+\ker A$ of the kernel. Over the real numbers, this gives exactly one solution when the kernel is zero and infinitely many when the kernel contains a nonzero vector: $x_0+tz$ is a distinct solution for each real $t$ if $z\ne 0$ lies in the kernel. An inconsistent linear system has an empty solution set. The dimension of a nonempty solution set, regarded as an affine space, is the nullity of $A$.