Question

Let W be a subspace of a vector space V over a field F. For any vV,the set {v}+W ={v+w|wW}
is called the coset of W containing v. It is customary to denotethis coset by v + W.

a) Prove that v+W is a subspace of V if and only if v W.

b) Prove that v1 +W =v2 +W if and only if v1 -v2 W.

Consider the collection, S = {v + W | v V } and define an additionon S by
(v1 +W)+(v2 +W)=(v1 +v2)+W for all v1, v2 V and scalarmultiplication by
c(v + W ) = cv + W
for all c F and v V.

c) Prove that the preceding operations are well-defined; that is,show that if
v1 +W =v1' +W andv2 +W =v2' +W as sets then

(v1 +W)+(v2 +W)=(v1' +W)+(v2' +W)

and c(v1 +W)=c(v1' +W) for all c F.

d) Prove that the set S is closed under the operations definedabove. In particular, S is a vector space called the quotientvector space of V modulo W and is denoted by V/W.