TITLE: Coset and set of all cosets
QUESTION [5 upvotes]: Let $C$ be a subspace of a vector space $V$ and $x \in V$.
The set
$x + C = \{ x + c : c \in C \}$ is called a coset of $C$.
The set of all cosets is denoted $V/C$.
Can you help me with some examples and intuition behind the concept of coset? 
I find hard to grasp the real meaning of this definition.

REPLY [5 votes]: I think it is important to remember that a coset is related with a equivalence class. In your case two vector are equivalent if their difference belongs to C. In this way, the coset is the set of all equivalent vectors. The intuition is that you are saying that all the elements become equal in their class. 
In R^2, when we have a line through the origin it can be represented by the origin itself. Taking all the parallel lines you now choose one point of each line to represent it. The set of coset can be associated with R^1 in this case.
I hope I could help.