Proving Linear Isomorphism: Quotient Spaces in Vector Subspaces

[lineintegral1]

Hey all,

We have not covered quotient vector spaces in class, but my homework (due before next lecture) covers a few proofs regarding quotient spaces. I've done some reading on them and some of their aspects, but as it is still a new concept, I am struggling with how to go about this proof.

Homework Statement

Let $M,N\subset L$ where $L$ is a vector space and $M,N$ are linear subspaces of $L$. Prove that the following mapping is a linear isomorphism:

$$(M+N)/N \rightarrow M/M \cap N$$2. The attempt at a solution

Well, I understand that the general approach to proving this would be to show that the morphism is bijective. Thus, I want to show that whatever is in the codomain is in the domain such that it gets mapped to the codomain. I also want to show that uniqueness in the codomain implies uniqueness in the domain. Can someone give me some tips as to how to work with quotient spaces such that I can relate these to more elementary linear algebra problems?

Hope that all makes sense. Thanks!

 

[Kreizhn]

This is actually a fairly generic results, and can be applied to groups, rings, modules, and vector spaces (although the notation has to change a bit). The way it is traditionally done is by finding a surjective map from M to $\frac{M+N}N$ and showing that the kernel is $M \cap N$. Can you think of a map for which this is true?

 

[lineintegral1]

Sorry, I'm still a little stuck here, are you just saying find any map with those properties? Is there a particular one I'm looking for? Or is it arbitrary? Again, I'm still trying to figure out how these quotient spaces work. How can I find such a map whose kernel relates to the intersection in that way?

 

[Kreizhn]

Maybe there is another way of doing this for vector spaces, and if that is the case then I'm sorry that I'm misleading you.

Anyway, you need to know two things. First off, is that if $\phi: V \to W$ is a linear map of vector spaces, then
$$\text{Im}\phi \cong V/\ker \phi$$
where $\text{Im}\phi = \phi(V)$ is the image of $\phi$.

Now since the map is linear, you know that $\phi(0_V) = 0_W$. Translating this into your language, you know that $\phi(0_M) = 0_{(M+N)/N} = N$. Hope that helps.