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.
The "real meaning" of the definition is the definition itself - but you're right to ask for examples and intuition.
First, geometry. A line $L$ through the origin in the plane $\mathbb{R}^2$ is a subspace. Any line parallel to $L$ is a coset of $L$. (You should be able to check that from the definition.) Then $\mathbb{R}^2/L$ is the set of all lines parallel to $L$. (Note that by convention $L$ is parallel to itself and thus a coset of itself.)
The only other subspaces of $\mathbb{R}^2$ are the trivial subspace $\{0\}$ and the whole space. The cosets of $\{0\}$ are all the points of the plane. The only coset of $\mathbb{R}^2$ is $\mathbb{R}^2$ itself.
In three dimensional space the cosets of a line (plane) through the origin is the set of all lines (planes) parallel to it.
Another way to gain some intuition for cosets is to think algebraically. Consider the linear equations $$ 2x + 3y = 0 $$ and $$ 2x + 3y = 15 . $$ The solutions to the first of these is the line through the origin containing the vector $(-3,2)$: all the pairs $(-3t, 2t)$. You can find all the solutions to the second equation by finding one, say $(6,1)$, and adding to it the general solution to the first: $(6-3t, 1+2t)$. That's just another way to say that the set of solutions to the second equation is a coset of the subspace of solutions to the first equation. If you started with a different single solution to the second equation, say $(0,5)$, you'd still have the same coset for all the solutions.
Clearly both the geometric and algebraic examples work in higher dimensions. They work too in more general mathematical structures you may encounter as you study more mathematics. In particular, watch for them in differential equations.
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.
First look at what the definition says (I am putting it in terms of groups rather than vector spaces, but the intuition is the same). It means that to define a coset $x+C$, you take an element $x \in V$ and iterate over all the elements of the subgroup $C$, to construct the coset $x+C$.
Consider the example of $\mathbb{R}/\mathbb{Z}$. If I take $x = 0.1$, then the (right) coset $x+\mathbb{Z} = \{..., -1+0.1, 0+0.1, 1+0.1, ...\}$. Now, you can notice that here the set of all cosets will essentially be the group (field as well) $\mathbb{R}$ itself because that set will have all the integers along with all possible real values, which is the set of all real numbers itself. But, you might argue that $\mathbb{R}/\mathbb{Z}$ is represented as the set $\mathbb{R}$ modulo 1. This is done by taking equivalence classes instead of the complete cosets, like in the example of the right coset $x+C$ above, we represent it as just the equivalence class [0.1]. This provides a succinct representation.
