# Correspondence between the projective space associated to a vector space and the dual space of the vector space?

Let $V$ be a vector space over $\mathbb{C}$ and $V^*$ be its dual space. Let $\mathbb{P}V$ be the projective associated to $V$. It is said that homogeneous coordinates of $\mathbb{P}V$ correspond to elements of $V^*$. What is the correspondence? Suppose that $V=\langle v_1, \ldots, v_k \rangle$ and $V^*=\langle v^*_1, \ldots, v^*_k \rangle$ where $v^*_i(v_j)=\delta_{ij}$. We can let $(a_0, \ldots, a_k) \in \mathbb{P}V$ corresponds to $f\in V^*$ where $f(v_i)=a_i$. Is this the correct correspondence? Thank you very much.

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The duality between $V$ and $V^*$ induces a "projective duality" between $\mathbb{P}V$ and $\mathbb{P}(V^*)$. For example, projective hyperlanes in $\mathbb{P}V$ correspond to vector hyperplanes in $V$. Such an hyperplane is the kernel of non zero linear form $f\in V^*$ unique up to a constant. In other words it corresponds to a line in $V^*$, i.e. a point in $\mathbb{P}(V^*)$. – AFK Sep 7 '11 at 17:04

Let $V$ be a finite dimensional $k$-vector space. A set of linear coordinates on $V$ is simply a basis $(\alpha_i)$ of $V^*$. Taking the coordinates of a vector $v\in V$ is considering its image under the isomorphism $\alpha: V \to k^n$, $v \mapsto (\alpha_1(v),\ldots,\alpha_n(v))$.
Now $\mathbb{P}(V) = (V\setminus \{0\})/k^\times$ and homogenous coordinates on $\mathbb{P}(V)$ are just linear coordinates on $V$ up to the action of $k^\times$. The map $\alpha: V \to k^n$ above induces $\bar{\alpha} : \mathbb{P}(V) \to (k^n\setminus \{0\})/k^\times = \mathbb{P}^n$. So a set of homogenous coordinates corresponds to a basis $(\alpha_i)$ of $V^*$ (and not just one element $f\in V^*$ as you seem to think).