 Left null space

For a real matrix $A$, its left null space consists of the vectors $w$ with $w^{T}A=0$, or equivalently $A^{T}w=0$. It is the null space of the matrix transpose, and is the orthogonal complement of the column space. The linear system $Ax=b$ is solvable exactly when $w^{T}b=0$ for every such $w$.