Rank of a matrix with affine columns
= Rank of a matrix with affine columns
If the $j$th column of a matrix is $v+c_j\mathbf1$, then its image lies in $\operatorname{span}\{v,\mathbf1\}$. If $v$ is not constant and at least two $c_j$ differ, the rank is exactly two; for $m$ columns the nullity is $m-2$.