1. ## proof

Let W be a subspace of a vector space V over a field F. For any v belonging to V the set {v} + W = {v+w: where w belongs to W} is called the coset of W containing v.

Prove that v+W is a subspace of V iff v belongs to W.

??

2. This is a simple idea.
$\displaystyle v \in W\quad \Rightarrow \quad v + W = W.$

If v+W is a subspace then $\displaystyle \left[ {\exists a \in W} \right]\left( {v + a = 0} \right).$

3. Originally Posted by ruprotein
Prove that v+W is a subspace of V iff v belongs to W.
Here is another exaplation.

If you studied group theory....
$\displaystyle aH=H$ of a subgroup $\displaystyle H$ or $\displaystyle G$ and $\displaystyle a\in G$.
If and only if $\displaystyle a\in H$.
The reason is as follows, the relation that defines cosets is an equivalence relation, hence it divides cosets into disjoint sets, and the set that contains $\displaystyle a$ is $\displaystyle aH$. Thus, because of disjointness we have that if $\displaystyle aH=H$ then $\displaystyle a\in H$.

Now by definition a vector space is a group (abelian) thus the same discussion in the preceding paragraph applies to a group as well.

4. how does by saying v + a = 0 say that v belongs to W

5. Isn't -a in W?

6. Originally Posted by ruprotein
how does by saying v + a = 0 say that v belongs to W
Because,
$\displaystyle v+a=0$
Thus,
$\displaystyle v=-a$
But if $\displaystyle a\in V$ then $\displaystyle -a\in V$ (one of the axioms for a vector space).
Thus,
$\displaystyle v\in V$