Solution
= Solution
Every column lies in $\operatorname{span}\{v,\mathbf1\}$. Conversely, differences of two columns are nonzero multiples of $\mathbf1$, while any one column together with $\mathbf1$ recovers $v$. Hence
$$
\boxed{\operatorname{im}A
=\operatorname{span}\{(1+b,2+b,3+b)^T,(1,1,1)^T\}},
$$
and
$$
\boxed{\operatorname{rank}A=2}.
$$
This is an instance of the <rank of a matrix with affine columns>.