Good day to all. I have begun covering the topic of linear transformations in a linear algebra course. I was was given a set of review exercices for an upcoming exam. For one of the questions, I am not sure if what I have done is correct. My doubts mostly stem from the use of homogeneous coordinates. Any advice-help would be greatly appreciated. The question and my work is attached as a pdf file.

Again, I am looking to validate my work and so I wish to immediately thank those taking the time to help me.