I remember that in my first linear algebra class, in section the GSI mentioned that the rank-nullity theorem was a consequence of the first isomorphism theorem. Now in my second linear algebra couse, the instructor used it to prove the RNT, but used the result: $$ \dim(V/U) = \dim V − \dim U $$
Earlier in the course, this theorem was proved using the RNT and the quotient map. Is there another way to prove the rank-nullity theorem from the first isomorphism theorem or does the validity of the statement rely on the fact there are other ways to prove the dimension of a quotient space?