confusion about cosets and quotient space The set of cosets $V/U = \{ v+U: u \in V \}$ with operations 
$$(v+U) + (w+U) = v+w+U$$
$$a(v+U) = av+U$$
(which is well defined) is a vector space called Quotient Space.

I am having a difficult time understanding what cosets really are. Is there a way to visualise it easily?
I was going over a proof that $\bar{B} = \{e+U : e \in B\backslash\mathcal{E}\}$ is a basis for $V/U$ when $\mathcal{E}$ is a basis of $U$ and $B$ is a basis of $V$ containing $\mathcal{E}$.
I understood part proving that it's a spanning set.
However in the part proving that it's linearly independent, I got confused. The lecturer wrote: 
"Assume for some $a_i \in \mathbb{F}$ and $e_i \in B\backslash\mathcal{E}$, 
$a_1 (e_1+U) + ... + a_n (e_n+U) = U$"
Why is it U and not zero on the RHS? is U = 0 in quotient space? If so, why?
Thanks. 
 A: To your last question, the "zero" in the vector space $V/U$ will be an element $e$ such that $q + e = q$ for all $q \in V/U$. You easily see that the role of zero here is played by $e = 0 + U = U$, as for each $v + U \in V/U$ one has
$$
(v + U) + (0 + U) = (v + 0) + U = v + U.
$$

As to visualizing cosets, take $V = \mathbb{R}^{2}$, and $U$ to be any $1$-dimensional subspace, that is, a line through $(0,0)$. Then the cosets of $U$ are nothing else but the lines parallel to $U$.
A: You should spend some time analyzing the definition of cosets and solving examples. This will help you 'visualize' the idea of cosets. Once you get the intuition it will be easy for you to prove the theorem. As your main concern is with understanding cosets I will focus on that in my answer.
First of all, remember that the set of cosets (under the operations you mentioned) is a group only when U is a normal subgroup of V.
Let's consider a group G and any subgroup H of G. Then for any element 'a' of G, the set aH = { ah | h is in G} is called the left coset of H in G containing a. Let's break up the definition: 'left' because we are multiplying the group element on the left hand side, 'containing a' because aH will definitely contain the element a (note that H is a subgroup so e belongs to H, and hence a = a.e belongs to H).
Now, for each element of G we will get a left coset of H in G. Note that the cosets may or may not be all distinct. When we collect all the left cosets together we get a set of all cosets, and when the subgroup H involved is Normal in G, these left cosets together form a group under the operations you mentioned. This group is referred to as the Quotient group or Factor Group. When you study Isomorphisms, you will see the significance of these groups.
An interesting thing to note here is that the cosets partition the group G, i.e, union of the distinct left cosets will give me all the elements of G. Now, pick up an example from your textbook and try and analyze these observations.
P.S. - I have spoken about 'left' cosets in the whole discussion. All the observations apply for 'right' cosets as well.
