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Obviously, a linear transformation over a space maps all straight lines to other straight lines.

My question is: is the converse true? That is, if we're looking at a space after some transformation and we're observing that all straight lines were mapped to straight lines, does that imply a linear transformation? (Lines are understood as finite and infinite collections of points that are aligned, well, in a line)

EDIT: The comments mention affine transformations which fullfil the above condition but are not linear since they also involve translation. I think I'll reask then: Is every transformation that maps ALL lines to lines necessarily affine?

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    Well, it's definitely false in $\Bbb R^1$. Linear transformations can be defined over all sorts of vector spaces over lots of different fields. Are you talking specifically about $\Bbb R^n$? – Greg Martin Sep 24 '14 at 06:56
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    No, it could be an affine transformation as well. – user_of_math Sep 24 '14 at 06:56
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    in $\mathbb{R}^2$ take a vector $(a,b)\neq 0$ then the map $(x,y)\mapsto (x+a,y+b)$ is certainly not linear. – Marc Bogaerts Sep 24 '14 at 09:21
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    A trickier question: in $\mathbb{R}^n$, for $n\ge2$, is every map which sends lines to lines and the point zero to zero linear? – Mike Earnest Sep 24 '14 at 15:28
  • @MikeEarnest I think I'll refine my question accordingly –  Sep 24 '14 at 19:02

1 Answers1

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not always but converse will be true iff T(x) that is a function of x .You can suppose y=T(x) represents the line that Passes through origin

Kislay Tripathi
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