Proving that the map $(M+N)/N\to M/(M\cap N)$, $(m+n)+N \mapsto m+(M\cap N)$ is a linear isomorphism

Question

$\textbf{1.}$ Let $M,N\subset L$. Prove that the following mapping is a linear isomorphism : $$(M+N)/N\to M/(M\cap N)\, ; \quad (m+n)+N \mapsto m+(M\cap N).$$

I really have no idea in questions like this. The question is on kostrikin's book (linear algebra and geometry).

Answer 1

I'll leave well-definedness and linearity to you to check. But, as an example, let's check injectivity.

Suppose $\phi(m_1 + n_1 + N) = \phi(m_2 + n_2 + N)$. Then $m_1 + M \cap N = m_2 + M \cap N$. Then $m_1 = m_2 + v$ where $v \in M \cap N$. We have $$ m_1 + n_1 + N = m_2 + v + n_1 + N $$ But, since $v + n_1, n_2 \in N$ we have $v+ n_1 + N = n_2 + N$. Hence $$ m_1 + n_1 + N = m_2 + n_2 + N $$ proving injectivity.

Note that if $L$ is finite dimensional, this suffices to prove surjectivity as well, so all there is left to check is well-definedness and linearity. Even if $L$ isn't finite dimensional, surjectivity follows almost immediately.

Answer 2

Consider first the map $$ f\colon M\to (M+N)/N,\qquad f(x)=x+N $$ This map is clearly surjective and its kernel is $M\cap N$. Thus the induced map $$ g\colon M/(M\cap N)\to (M+N)/N,\qquad g(x+M\cap N)=x+N $$ is an isomorphism.

Its inverse map is the one you are given.