A transformation transforms a line to a line, if and only if it is a linear transformation

Asked Dec 06 '19 at 00:34

Active Dec 06 '19 at 00:34

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Given $T(x,y)=(x',y')$ prove that the following two conditions are equvalent:

T transforms straight lines into straight lines.

T is a linear transformation.

The "if" direction is easy. However, I have no clue how to start with the only if direction. Probability I would prove by contradiction by assuming that, if a transformation is not linear, then it transforms line to other things?

edited Jun 12 '20 at 10:38

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asked Dec 06 '19 at 00:34

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Does this answer your question? How to prove that linear transformations transforms lines into lines – Dec 06 '19 at 00:46
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Here's a counterexample: $T(x,y)=(x+1,y)$ transforms straight lines into straight lines, but $T$ is not a linear transformation. – Lee Mosher Dec 06 '19 at 00:55
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See my old question. You will need some more assumptions; at the very least you must assume that $T(0) = 0$. If we are also given that $T$ is bijective, then this is a consequence of the "fundamental theorem of projective geometry". – Ben Grossmann Dec 06 '19 at 10:27
See also this question – Ben Grossmann Dec 06 '19 at 10:30
To flesh out the comment by @Omnomnomnom, some linear transformations map some lines to a point (the origin), so even the easy "if" direction needs "more assumptions". – hardmath Dec 17 '19 at 01:51

A transformation transforms a line to a line, if and only if it is a linear transformation

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