Motivation and Meaning of a Quotient Space

Question

Let $V$ be some space and $W$ a subspace, we usually define the quotient space as $V/W$. If $V$ is a group under $+$ and $W$ a subgroup, then we write $V/W = [ {v+W| v \in V }] $ what is the motivation behind this and what is the meaning of this? Or let me put it this way, how to visualise the quotient space intuitively? Or maybe in layman terms.

Answer 1

The quotient space partitions your group. Meaning it divides/subsections your group into subsets all of the same order -- known as cosets.

Consider the integers $\mathbb{Z}$ let your subgroup be the multiples of three, that is: $$ \{z \mid z=3n \ \ \text{for some}\ \ n \in \mathbb{Z} \} $$

then $13=1+3 \cdot 4$ and $22=1+3 \cdot 7$ both belong to the same coset. Since they are both of the form $1+3n$. similarly, all numbers of the form $3n$ and $3n+2$ belong in the same coset. We have just taken the integers, based on your definition of a quotient space, and divided/partitioned it into subsets