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I want to start with definitions. Affine space $\mathbb A^n$ over the field $K$ is the set of $x_i$'s i.e., $\mathbb{A}^n = \{(x_1,\dots,x_n):x_i \in K \}$

if $n=2$, projective line defined to be $\mathbb P^2$ is set containing all lines through the origin in $\mathbb A$. Here, explanation was link, Points of the projective line are the lines through the origin in $\mathbb R^2$.

Why do we get the right to think this over $\mathbb R^2$ even though our entries coming from the field $K$? Of course, if we particularly choose space to be projective real line then I am okay with that. What strange to me is many sources explain sort of classic concept. I mean draw all lines passing through origin and they must meet at $x=1$. They all think the concept on $\mathbb R$, what about any field $K$. I can't even imagine how can I draw lines through the origin in arbitrary field $K$?

Elise9
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  • Concerning lines for arbitrary $K$ see this post, for example. – Dietrich Burde Apr 15 '24 at 20:39
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    Even if you can't imagine drawing the line, the concept of a line still makes sense very naturally. Hilbert's quote: "One must be able to say at all times--instead of points, straight lines, and planes--tables, chairs, and beer mugs". We have a good drawing model in the case of real field, and it helps our intuition for more cases. – okabe rintarou Apr 16 '24 at 07:29

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You should think of the picture of the real case an analogy. Some of the properties go across the analogy (eg two distinct lines through the origin determine a unique plane containing both) and some of them might not (eg each plane through the origin contains infinitely many lines through the origin over $\mathbf{R}$ but not over a finite $K$). Knowing whether some specific fact can be checked by analogy requires experience.

If your definition of the projective plane (space) is the set of lines through the origin then a reasonable definition of a projective line that does not require this analogy is (the set of lines contained in) a plane through the origin. But the reason it's called a line is ultimately because of the analogy.

To be precise, by a line (resp. plane) in $K^n$ or $\mathbb{A}^n(K)$ I mean a $1$-dimensional (resp. $2$-dimensional) linear/affine subspace, ie one cut out by $n - 1$ (resp. $n - 2$) independent linear equations.

ronno
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