To me it's not clear exactly what you are asking.
Are you asking: If is a linear map between vector spaces and with , show that need not be an isomorphism?
If so, just let be the linear map sending all elements of to the zero vector of (which won't be an isomorphism, whenever ).
And why should the equations imply that ? The map could easily have rank 1, and thus non-zero nullity (at least as far as ).
About non-linear maps not being isomorphisms of vector spaces: Well, yeah, it is very much a true statement, but I really think that the map is implicitly assumed to be linear, so the author of the problem would probably like a linear example of a non-isomorphism.