1. Given a vector space V and a subspace W of V.
  2. "x≡y (mod W) if x-y∈W" defines equivalence relation.
  3. equivalence relation partitions V into equivalence classes.
  4. equivalence classes is called cosets of W in V.
  5. Define the quotient space V/W, whose elements are cosets of W in V.

My question:

If x & y ∈ W, x-y∈W. So it seems W⊆quotient space since W is a coset of itself. True? If yes, why V/W?


1 Answer 1


There is an element of the quotient space that has the same members as $W$, so $W$ is an element of $V/W$, but no, $W$ is not a subset of the quotient space. A subset of the quotient space would be a collection whose members are equivalence classes.

You are right when you say that $W$ is a coset of itself though, and that is why $W$ is an element of $V/W$.

  • $\begingroup$ Just ask a short question: to V/W, what does / mean? exclusion? $\endgroup$ Sep 13, 2014 at 20:13
  • 1
    $\begingroup$ @sleevechen $/$ merely indicates that this is a quotient space. In general $/$ is used to denote a quotient structure, where the object on the left is the base structure and the object on the right is what cosets are being formed from. Sometimes an equivalence relation will appear on the right when it is more convenient to think in terms of equivalence classes instead of cosets. The notation is sometimes read as "$V$ mod $W$." $\endgroup$
    – Benjamin
    Sep 13, 2014 at 21:50

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