I'm currently reading about quotient spaces in the context of linear algebra. I found the description on Wikipedia helpful. In particular I liked the example of two subspaces $V, W$ of $R^{2}$, where one could define $W$'s basis as a single vector, say $e_{1}=(1,0)$, thereby defining a defining the quotient space $V/W$ whose equivalence relationship captures lines whose difference is a vertical line. The quotient space is then the set of all cosets ${x+n: n\in W}$ for all $x\in V$. In this way one can build a description of all the vectors parallel to $e_{1}$ in terms of there sum with members of $W$. Since $W$ is a vector space, it seems like it must have a lower dimension than $V$ to generate a quotient space that isn't equal to $V$. Since $V/W$ is read "$V$ mod $W$" I was immediately reminded of the modulo operator in various programming languages. I hoped to apply the previous concept to a set of lower dimension.
The concepts appear related, and I think you could construct an analogous structure as follows. Given a set that was closed under addition, and whose 'basis' was a single number say $N={3}$ one could construct a space that represented the modulus of the integers by $3$. This appears analogous to the way in which one space must have lower dimension than the other in the previous example. Furthermore I think that cosets defined for $\{0,1,2\}$ would contains all the other cosets, yielding three equivalence classes. However, neither of these sets is closed under scalar multiplication, they aren't vector spaces. Is this an acceptable construction? It doesn't seem like it would be correct to call it a quotient space, but it seems to describe something useful. If this is a valid construction are there further properties or operations of interest? If this is not a valid construction, is there a way I could go about rigorously constructing one?
I apologize if this is a duplicate question, I don't totally understand how to describe what I'm interested in, and as such probably missed relevant information.
Best Answer
Actually, what you defined is a quotient space: it's the quotient of $\mathbb Z$ by the equivalence relation “congruent modulo $3$”.
And the quotient of a vector space $V$ by a vector subspace $W$ is also the quotient of a set ($V$) by an equivalence relation ($v\sim v^\star\iff v-v^\star\in W$). So, yes, it is quite similar.