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http://en.m.wikipedia.org/wiki/Quotient_space_(linear_algebra)#section_4

(I'm trying to understand the definition on wikipedia.)

Let $W$ be a subspace of a vector space $V$.

Wikipedia defines an equivalence relation $\sim$ on $V$ as "$x\sim y$ iff $x-y\in W$", so that $[x]=\{y\in V:x-y\in W\}$. Then immediately wikipedia states that $[x]=x+W$.

I'm completely confused. If $[x]$ is an equivalence class of a relation $\sim$ defined as above, it's clear that $[x]\neq x+W$.

Shouldn't it rather state that $[x]+W\triangleq x+W$?

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No, it is not clear that $[x]\neq x+W$, since it is true that these sets are equal. –  1015 Feb 9 '13 at 15:32

2 Answers 2

$[x]$ is a subset of $V$, namely $$[x]=\{y\in V\mid y\sim x\}.$$ $x+W$ is also a subset of $V$, namely $$x+W=\{x+w\mid w\in W\}.$$ These sets are the same. Two remarks: $x+W$ is in fact a common abbreviation for $\{x\}+W$. And $[x]+W$ is indeed the same, but only because $[x]+W=[x]$.

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The equivalence class of $x$, denoted by $[x]$, is a subset of $V$ consisting of all elements of $V$ which are related with $x$ by the relation $\sim$.

Furthermore the subset $\{x+w; w\in W\}$, denoted by $x+W$, consist of all elements $y\in V$ such that there exist $w\in W$ with $y=x+w$, that is, $y-x=w\in W$ (or $x-y=-w\in W$ also).

So those two subsets are the same.

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