Do these constants stay not zero when taking the negation of the consequent in the contraposition?

Question

I was told to prove that if $v_{1},...,v_{k}$ is a linearly independent list of vectors in $V$, then $c_{1}v_{1},...,c_{n}v_{n}$ are linearly dependent, with $c_{i}\not=0$ for all $i\in\{1,...,k\}$. I was planning on proving it by contraposition; however, I'm wondering whether the $c_{i}$'s stay not zero when I take the negation of the consequent.