Suppose I have a map on 3D space,such that it maps any line to a line or a point and maps origin to origin. Can I conclude that T is a linear map? If not, could you provide a counterexample and if yes, then could you provide a proof? I think it will help a lot to visualize the transformations geometrically.
2026-04-08 00:46:22.1775609182
T is a map on $\mathbb{R}^3$ and $T$ maps every line either to a line or a point and sends $(0,0,0)$ to $(0,0,0).$ Is it a linear map necessarily?
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Consider $T(x,y,z) = (x^3,0,0)$.