Your proof does not make sense because T(x) is not defined.
For the second part.
I have to either show that a transformation is either linear or not due to the additivity or the homogeneity.
The first question is:
m(x+1)+my= mx+m +my
mx+my =/= mx+m+my
T(x,y)=(x+1,y) is not linear under additivity.
How would I prove or disprove that the transformation is linear under homogeneity?
A second question is:
For this, I would have:
T(\vec a) + T(\vec b)=?
I'm not sure what needs to be done here...
For the second part, I'm not sure how to do what you did above using two different vectors, because I would be comparing x to y at some point, correct?
Sorry for my idiocy, by the way.