Quotient Space Zero Element

Question

Im studying quotient spaces for the first time, and I am doing a small exercise in which I am asked to show the the quotient space is a vector space. I am not asking about the exercise itself, but a question came up when I was completing it: in a vector space we must have an additive inverse and an a zero element, and they should be unique, but I curious that if we have a vector space $V$ and a subspace $W$, and we define the coset of $v \in V$ and $W$ as: $v + W = \{v+w | w \in W\}$, then defining addition such that $(v+W) + (u+W) = (v+u) + W$, we can show that if $w_1 \in W$:

$$w_1 + W = W \rightarrow (v+w_1) + W = v + W$$

Do we then still consider the zero vector as unique since we speak of the quotient space as the space of objects $v+W$? So then $w_1 + W$ and $W$ being the same objects, there is no contradiction? Thanks in advance!

Answer 1

In the quotient space of $V$ and $W$ the elements (or 'vectors') are equivalence classes of elements of $V$. As you already pointed out, for $w\in W$ we have

$w+W = W = 0+W$,

so in this case, $w$ and $0$ are in the same equivalence class, and therefore are considered the same object in the quotient space. Therefore, we do not get any problems.

Answer 2

If $w\in W$ then $W$ and $w+W$ are exactly the same coset, i.e the same element of $V/W$. You can indeed describe this element in different ways, but it is still just one element.

Just like in $\mathbb{R}$ you can represent $0$ in different ways: $0, \ \ 1-1, \ \ 2+3-4-1$ and so on. It is still the unique zero element of $\mathbb{R}$.