Left null space ($\ker A^T$)

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 $w$.

For the linear ordinary differential equation $c^{\prime }=Ac-b$, a fixed left null space vector gives $(w^{T}c{)}^{\prime }=-w^{T}b$. Thus $w^{T}c$ is a conserved quantity when $w^{T}b=0$. In a box model written for concentrations per unit volume, physical tracer inventory uses the box volumes as weights. Zero column sums make $w=1$ possible, but that property must be checked rather than assumed.