Let W be a subspace of a vector space V over a field F. For any v V the set {v} + W = {v

QUESTION:

Let W be a subspace of a vector space V over a field F. For any v V the set {v} + W = {v + w: w W} is called the coset of W containing v. It is customary to denote this coset. by v + W rather than {v} + W. (a) Prove that v + W is a subspace of V if and only if v W. (b) Prove that c, + W = v2 + W if and only if vx - v2 W. Addition and scalar multiplication by scalars of F can be defined in the collection S = [v f W: v V} of all cosets of W as follows: (vi + W) + (v2 + W) = (Vj + v2) + W for all v\,v2 V and for all v V and a F. a(v + W) = av + W (c) Prove that the preceding operations are well defined; that is, show that if vi + W = v[ + W and v2 + W = v'2 + W, then and (c, + W) + (v2 + W) = (v[ + W) + (v'2 + W) a(Vl + W) = a(v[ + W) (d) for all a F. Prove that the set S is a vector space with the operations defined in (c). This vector space is called the quotient space of V modulo W and is denoted by V/W.