Cosets in R3 Using the following definition, in Problems 1 and 2, find the W-cosets for the give vectors v̅ and given a graphical description of each?
Cosets
If W is a subspace of vector space that includes the origin, and; v̅ is a vector in Rn, then the W-coset of v̅, denoted v̅ + W, is the set defined by
v̅ + W = [v̅ + w̅ | w̅ is in W)
Thus, in R3, if W is a plane passing through the origin, then the coset v̅ + W is a plane parallel to W passing through v̅?
1. W = {[x1, x2, x3] | x1 + x2 + x3 = 0}; v̅ = (0, 0, 1)
2. W = {[x1, x2, x3] | x3 = 0};v̅ = (1, 1, 1)