= Image obstruction by an annihilating functional
{title2=$n^TA=0,\ n^Tb\ne0$}
If a <linear functional> $\ell$ vanishes on the <image of a linear map> $T$ but $\ell(b)\ne0$, the equation $Tx=b$ has no solution. In Euclidean coordinates, a vector $n$ with $n^TA=0$ provides such an obstruction whenever $n^Tb\ne0$. In finite dimensions these annihilating functionals exactly characterize membership in the image: $b\in\operatorname{im}A$ precisely when it is orthogonal to every vector in $\ker A^T$.
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