Cosets - [·]

Cosets - Examples

Vectors

The elements (vectors) of a vector space form an abelian group under vector addition. The subspaces of the vector space are subgroups of this group. For a vector space 𝑉, a subspace 𝑆, and a fixed vector 𝑣∊𝑉, the set

1. [𝑣] = 𝑣 + 𝑆
2. [𝑣] = {𝑣+𝑠 : 𝑠∊𝑆}
3. [𝑣] = {𝑥∈𝑉 : 𝑥=𝑣+𝑠, 𝑠∈𝑆 } # all three are equivalent

is called an affine subspace, and is a coset of 𝑣 often denoted as [𝑣] (both left and right, since the group is abelian).

Example

If:

• 𝑆 = { (𝑥,2𝑥)∊ℝ² : 𝑥∊ℝ }
• 𝑣∊𝑉

Then the coset of 𝑣 is parallel to 𝑆

• [𝑣] = { 𝑥∈𝑉 ∣ 𝑥 = 𝑣+𝑠, 𝑠∈𝑆 }

coset.drawio

Resources

• https://en.wikipedia.org/wiki/Coset