I'm following the proof of the fact that $ \mathrm{dim}(U) + \mathrm{dim} \mathrm{Ann}(U) = \mathrm{dim}(V) $ for $U \subset V$ and $\mathrm{Ann}(U)$ is an annihilator of $U$, in here (Proposition 2.20 (a)).
But I don't understand how $v^{'}(v_{i}) = c_{i}$.