Let $A$ be a nonsingular $n\times n$ matrix, $B= \begin{bmatrix} c_1&c_2&\dots&c_m\end{bmatrix}$ is an $n\times m$ matrix written in terms of its column and $v$ is a column vector of length $m$.
If $||ABv||<M$, does it imply that each vector $c_i$ has a vector norm at most $M$?
I solved that $ABv$ is a 'linear combination' of $Ac_i$'s but I don't know if that will help.