# Quotient Space Zero Element

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!

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.
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 $$\mathbb{R}$$.