Give an example of a linear transformation T : R^2 --> R^2 such that N(T)=R(T)?

Question

and an explanation would be much appreciated.

Answer 1

Is N(T) the nullity? Is R(T) the rank? If so, I think if you define T so that it maps R^2 onto a 1-dimensional image-space, you may have what you need. Will T(x,y) = x + y do it?