im trying to understand the proof on rank (fg) = rank (f) if g is isomorphic.
it states that since g is surjective, it means that im(f)= im (fg)..but i dont get how it means that im(f)= im (fg)
To prove that two sets are equal, prove that each is a subset of the other. That is, prove that "if x in A then it is in B" and vice versa. Here "A" and "B" are im(f) and im(fg).
If vector w is in Im(f) then there exist vector v in V such that f(v)= w. Since g is isomorphic, there exist vector u in U such that g(u)= v. Then fg(u)= f(v)= w so w is in im(fg).
Now do it the other way around: if vector w is in Im(fg) then there exist vector u in U such that fg(u)= w. Let v= g(u). Then f(v)= w so w is in im(f).