Vector space and its Projectivized Space

Question

Why is the co-dimension one subspaces are the points of $\mathbb P(V^{\vee})$.

$V^{\vee}$ is the dual space of V and and $\mathbb P(V)$ is the projectivized space of V.

$\mathbb P(V)= \frac{V-\{0\}}{\sim}$ where $v_1 \sim v_2$ iff $\exists\lambda\in k$ s.t. $v_1=\lambda v_2$

Answer 1

Codimension one subspaces are kernels of linear functionals uniquely defined up to scalars.