= Solution set of a linear system
{title2=$\{x:Ax=b\}=x_0+\ker A$}
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 $z\ne0$ 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 $A$.
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