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Let $W \subset V$ be vector spaces. I don't understand the quotient space $V/W$. I read the Wikipedia and searched this site.

I would have thought: say the vector space operation is $+$. let $Q = V/W$. Then $V = W+Q$ by "multiplying across". So $Q$ contains elements of the form $V + (-1)W$. Why isn't this how the quotient space is defined?

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    You have $Q = \{ \{v\}+W | v \in V \}$. I suspect the term quotient came from group notation?2012-10-21
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    A Vector space is an abelian group! So the quotient space would contain the cosets of $W$!2012-10-21
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    You can't "solve" equations for sets. If $A=B+C$ as sets, that doesn't imply that $C=A-B$. (Take $A=B$ to be any set and $C=\{0\}$, for example.)2012-10-21

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