# Projective plane and its dual

So the projective plane $\mathbb{RP}^2$ is not a vector space. Is it still isomorphic to its dual? If not, is there at least an invertible map that takes $\mathbb{RP}^2$ to its dual?

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My understanding of the term "dual" in projective geometry is that it only makes sense in an ambient projective space. What definition of "dual" are you using here? –  Qiaochu Yuan Jul 28 '10 at 1:59

Yes, though the word "dual" is somewhat questionable. If you mean is $\mathbb{P}(\mathbb{R}^3)$, the standard projective plane, isomorphic to $\mathbb{P}((\mathbb{R}^3)^*)$, the projectivization of the dual, then yes, it follows from the isomorphism of vector spaces.

Much more interestingly, the duality allows you to switch points and lines in theorems, such as the Mystic Hexagon and Brianchon's Theorem.

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Yeah, I'm referring to $P(\mathbb{R}^3)$ and $P((\mathbb{R}^3)^*)$. So how do you define the isomorphism? –  Adeel Jul 28 '10 at 2:06
You just take the isomorphism from to that you get by picking a basis, and that gives you an isomorphism (say, as smooth manifolds or real algebraic varieties). To clarify slightly, it induces an isomorphism by just applying it to the homogeneous coordinates. –  Charles Siegel Jul 28 '10 at 2:14