I have been given a 2-part question which first states given a vector space (V,K) and W$\subseteq$V is a subspace. that a W-affine subspace S$\subseteq$V is one in which s,s' $\in$ S, s-s' $\in$ W and $\forall$s$\in$S, $\forall$w$\in$W s+w$\in$S.
I had to prove ,given sets S, T $\subseteq$V were W affine subspaces of V and c$\in$$\boldsymbol{R}$, that S+T and cT are again W-affine subspaces of V.
It was given that S+T= {s+t|s$\in$S,t$\in$T} and cT= {ct|t$\in$T if c$\not$=0 and W if c=0}.
I solved this by saying:
Aim: to show that $\forall$(s+t),(s+t)' $\in$(S+T), (s+t)-(s+t)'$\in$W and (s+t)+w$\in$(S+T).
- S+T= {s'+t'=(s+t)'|s'$\in$S,t'$\in$T}
$\Rightarrow$ (s+t),(s+t)' $\in$ (S+T)
- s-s'=0$_v$ $\in$W, Since s+w $\in$ S
$\Rightarrow$ S+T= {s+w+t+w| (s+w)$\in$S,(t+w)$\in$T}
={s+s-s'+t+t-t'=(s+t)-(s+t)'+(s+t)} but [(s+t)-(s+t)' $\in$ W]
$\Rightarrow$(S+T)+W $\in$ (S+T) .
Now addressing cT=ct|t$\in$T if c $\not$=0 and W if c=0
we know t-t'=0$_v$ $\in$W , Also t+w $\in$T $\Rightarrow$ t+0$_v$ $\in$T
Because T is a subspace of V we know $\lambda$T$\in$T|$\lambda$$\in$K
$\Rightarrow$c=$\lambda$. Also when c=0=$\lambda$: ct=0t=0$_v$ $\in$ W.
Therefore S+T is an affine subspace since we proved our aim.
The problem I am facing occurs when i move to the next step which asks me to show using the operations that turn the set of all W-affine subspaces into a vector space that will denote the symbol V/W and be called the quotient space.Noting that all the vectors in the quotient space are W-affine subsets of V.
I have an idea how to do it but I am not sure if I am correct. I am thinking that addition and scalar multiplication that was used in the previous questions makes the vector space but I am not sure why this is the quotient space,so I am not sure if i am correct.