Homogeneous linear system
= Homogeneous linear system
{title2=$Ax=0$}
A <linear system> is homogeneous when its forcing is zero. Its solution set is the <kernel of a linear map>, a <vector subspace>. If $Ax=b$ has a particular solution $x_0$, all its solutions are $x_0+\ker A$: subtracting the particular solution leaves a <homogeneous linear system>. The <rank-nullity theorem> gives the dimension of this freedom.