As the title, I would like to know: Are projective transformation linear and why?
I'm talking about projective transformation, linear transformation in terms of homogeneous coordinates.
I can prove that in this context, all linear transformations are projective. But I cannot verify the opposite statement.
Could you guys please help me?
Yes and no. If by "projective transformation" you mean any collineation of the projective space, they can all be obtained by composing a linear map with an automorphism (this is the Fundamental Theorem of Projective Geometry), so they are not necessarily linear. If you are specifically referring to homographies, then they are always linear.