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How do I show that the triples $(𝑎, 𝑏, 𝑐)$ and $(𝑎′, 𝑏′, 𝑐′)$ define the same projective line if and only if $(𝑎, 𝑏, 𝑐) = 𝜆(𝑎′, 𝑏′, 𝑐′)$ for some nonzero $𝜆 ∈ ℂ$

I've seen that this is the definition of a projective plane, but I don't know how to actyally prove this. Any help would be great!!

Edit: A projective line in $ℙ^2(ℂ)$ is defined by $𝑎 𝑥 + 𝑏 𝑦 + 𝑐 𝑧 = 0$ where $0 ≠ (𝑎, 𝑏, 𝑐) ∈ ℂ^3$

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    $\begingroup$ What definition of $\mathbb{C}P^2$ are you using? Usually this is just the definition of the defining equivalence relation on $\mathbb{C}^3$. $\endgroup$ – Nate Gallup Apr 8 at 2:47
  • $\begingroup$ I added the definition in the post! $\endgroup$ – user65432109 Apr 8 at 2:58
  • $\begingroup$ Projective lines are just equivalence classes of hyperplanes in $\mathbb{C}^3$. $\endgroup$ – mathmath Apr 8 at 3:00
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This is straightforward. You show that any point $(x,y,z)$ on the projective line $(a,b,c)$ is also on the projective line $(a’,b’,c’)$; and you also show the converse i.e. that any point $(x,y,z)$ on line $(a’,b’,c’)$ is also on line $(a,b,c)$.

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