Thread: How to show this isomorphism?

Given . How can I show that the qutioent space is isomorphic to im A? Thank you very much!

what is the Group ??

L(V, W) represents all linear transformations from V to W

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).

Yes, sorry I meant that

Originally Posted by gotmejerry

Given . How can I show that the qutioent space is isomorphic to im A? Thank you very much!

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).