Given . How can I show that the qutioent space is isomorphic to im A? Thank you very much!
Then A is a linear transformation while V/ker(A) is a vector space. What do you mean by an isomorphism between a linear transformation and a vector space?
I suspect you mean an isomorphism between the image of V under A and V/ker(A).
This is just the first isomorphism theorem for vector spaces. Try showing that is a well-defined isomorphism. Now (and I mention this only because [despite the strangeness] some classes do this FIRST) if you know the rank-nullity theorem and you're in finite dimensions you know that and so and so the isomorphism follows.
if you start with a basis , of ker(A) you can extend this to a basis for V.
prove that is a basis for V/ker(A), and that
is a basis for im(A) (this proves the rank-nullity theorem, and gives you drexel28's isomorphism, at the same time).