This might be a very silly question, but we know that $\mathbb{R}\mathbb{P}^n$ is defined as the equivalence classes $[x]$ ($x\in\mathbb{R}^{n+1}$ nonzero) such that $x \sim y \iff x = \lambda y$ for any real, nonzero $\lambda$.

So, these elements in $\mathbb{R}\mathbb{P}^n$ are lines passing through the origin but without the origin in the line.

My question/problem comes with Lee's definition of $\mathbb{R}\mathbb{P}^n$ as the collection of $1$-dimensional linear subspaces of $\mathbb{R}^{n+1}$. But through this definition, it would make sense that the origin would be included in every equivalence class of $\mathbb{R}\mathbb{P}^n$ since $\textbf{0} \in \text{span}(x)$ for any $x\in\mathbb{R}^{n+1}$.

Why do these two definitions not seem equivalent? What am I missing?

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    $\begingroup$ The "origin" is not a one-dimensional linear subspace of $\Bbb R^{n+1}$. $\endgroup$ – Lord Shark the Unknown May 2 '19 at 6:53
  • $\begingroup$ @LordSharktheUnknown Yes, I understand that we don't take the origin to it's span since it's span is zero dimensional. But, I guess where the main confusion comes from, is that the span of any set with one vector will include the zero vector (origin). Edit: I'm an idiot. Never mind, thank you. $\endgroup$ – Good Morning Captain May 2 '19 at 6:59

You are right that these two definitions are not the same. But there is a canonical bijection between the two; simply add the origin to each equivalence class to get Lee's definition. In fact this is an isomorphism between the two different constructions of real projective space.

Just as there are many different constructions of $\Bbb{R}$, there are many different constructions of $\Bbb{RP}^n$. But the underlying construction is not relevant (most of the time). What matters is that they are isomorphic as projective spaces.


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