# How to identify the homogeneous coordinates on $\mathbb{P}V$ with the elements of $V^*$?

Let $V=K^{n+1}$ be a vector space of dimension $n+1$ and $\mathbb{P}V$ the projective space associated to $V$. How to identify the homogeneous coordinates on $\mathbb{P}V$ with the elements of $V^*$? Thank you very much. I think that the homogeneous coordinates on $\mathbb{P}V$ are $Z_0, \ldots, Z_n$, where $Z_0, \ldots, Z_n$ are standard coordinates on $K^{n+1}$.

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As you wrote, $Z_i$ is a standard coordinate on $K^{n+1}$, in other words a linear map $Z_i: K^{n+1}\to K$, in other words an element of $V^*$ where $V=K^{n+1}$.
Precisely. And those are nothing but (the standard basis) elements of $V^*$.