# 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.

-

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 $M/M\cap N$ – Pilot 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? – Pilot Dec 4 '12 at 5:50
Yes, which can be written as $m+N$. – Jonas Meyer Dec 4 '12 at 5:51
show 1 more comment

## 2 Answers

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.

-

For some basic information about writing math at this site see e.g. here, here, here and here. – Julian Kuelshammer Dec 18 '12 at 22:51
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 22, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9437313675880432, "perplexity_flag": "middle"}