Editing Coset (section)

=== Vectors ===
Another example of a coset comes from the theory of [[vector space]]s. The elements (vectors) of a vector space form an [[abelian group]] under [[vector addition]]. The [[linear subspace|subspaces]] of the vector space are [[subgroups]] of this group. For a vector space {{math|''V''}}, a subspace {{math|''W''}}, and a fixed vector {{math|'''a'''}} in {{math|''V''}}, the sets
<math display="block">\{\mathbf{x} \in V \mid \mathbf{x} = \mathbf{a} + \mathbf{w}, \mathbf{w} \in W\}</math>
are called [[affine subspace]]s, and are cosets (both left and right, since the group is abelian). In terms of 3-dimensional [[Euclidean space|geometric]] vectors, these affine subspaces are all the "lines" or "planes" [[Parallel (geometry)|parallel]] to the subspace, which is a line or plane going through the origin. For example, consider the [[Plane (geometry)|plane]] {{math|'''R'''<sup>2</sup>}}. If {{mvar|m}} is a line through the origin {{mvar|O}}, then {{mvar|m}} is a subgroup of the abelian group {{math|'''R'''<sup>2</sup>}}. If {{mvar|P}} is in {{math|'''R'''<sup>2</sup>}}, then the coset {{math|''P'' + ''m''}} is a line {{math|''m''′}} parallel to {{mvar|m}} and passing through {{mvar|P}}.<ref>{{harvnb|Rotman|2006|loc=p. 155}}</ref>