Suppose I have a $N \times N$ matrix $A$ which is a projection matrix on the column space of another matrix $B=[b_1, \ldots, b_n], n<N $, that is, $A=BB^\dagger$, with $^\dagger$ denoting the pseudoinverse. Clearly every vector $b_i$ belongs to the column space of $B$ and as such $A b_i= b_i$.
I want to modify matrix $A$ in such a way that $b_j$, for a specific fixed $j$, does not belong anymore to the subspace spanned by $A$. So I would like $A b_j \neq b_j$. How to achieve it?