= Orthogonality to the image of a matrix
{title2=$n^TAr=0\ \forall r\iff A^Tn=0$}
For a real <matrix> $A$ and a fixed <vector> $n$, every <vector> in the <image of a linear map> defined by $A$ is <orthogonal> to $n$ exactly when $A^Tn=0$. Indeed, $n^TAr=(A^Tn)^Tr$. If this vanishes for every $r$, choose $r=A^Tn$ to obtain $|A^Tn|^2=0$. Equivalently each column of $A$ is <orthogonal> to $n$. Requiring the statement only for nonzero $r$ changes nothing, since the zero input already has zero <dot product>. This is the <orthogonal complement> relation between the <image of a linear map> and the <kernel of a linear map> defined by its <transpose>.
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