# Isomorphism between 2 quotient spaces

Let $M,N$be linear subspaces $L$ then how can we prove that the following map $$(M+N)/N\to M/M\cap N$$ defined by $$m+n+N\mapsto m+M\cap N$$ is surjective? Originally, I need to prove that this map is bijection but I have already proven that this map is injective and well defined,but having hard time to prove surjectivity,please help.

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What does a typical element of $M/(M\cap N)$ look like; i.e., how do you write its general form? Given such an element, can you find an element of $(M+N)/N$ that is sent there under your map? Incidentally, note that $n+N=N$, so $m+n+N=m+N$ (assuming here that $n\in N$). – Jonas Meyer Dec 4 '12 at 5:43
the typical element is shown above,well your 2nd question is obviously equivalent to the surjectivity of f,i.e what I need!So is there an element that would preimage of arbitrarily taken element of $M/M\cap N$ – p.s Dec 4 '12 at 5:46
Given $x=m+M\cap N$, what element of $(M+N)/N$ might go to $x$? – Jonas Meyer Dec 4 '12 at 5:47
$m+n+N$ with n being arbitrary vector in N? – p.s Dec 4 '12 at 5:50
Yes, which can be written as $m+N$. – Jonas Meyer Dec 4 '12 at 5:51

Given an arbitrary element $x=m+M\cap N$ of $M/(M\cap N)$, note that $m+N\in (M+N)/N$ is mapped to $x$.
Define $T: M \to (M+N)/N$ by $m \mapsto m+N$. Show that it is linear and onto. Check the $\ker T$ and is $M\cap N$ by the first isomorphism theorem $f:M/(M\cap N) \to (M+N)/N$ is an isomorphism.