Let W be a subspace of a vector space V over a field F. For any v belonging to V the set {v} + W = {v+w: where w belongs to W} is called the coset of W containing v.
Prove that v+W is a subspace of V iff v belongs to W.
??
Here is another exaplation.
If you studied group theory....
of a subgroup or and .
If and only if .
The reason is as follows, the relation that defines cosets is an equivalence relation, hence it divides cosets into disjoint sets, and the set that contains is . Thus, because of disjointness we have that if then .
Now by definition a vector space is a group (abelian) thus the same discussion in the preceding paragraph applies to a group as well.