Question about proof on isomorphism theorem 
Let $U$ and $W$ be subspaces of a vector space $V$. Prove that the quotient spaces $$(U+W)/U \ \ \text{and}\ \ W/(U\cap W)$$ are isomorphic.

Proof :
The map
$$
\alpha\colon W\to (U+W)/U
\qquad
\alpha(x)=x+U
$$
is a linear map. Its kernel is $U\cap W$ and therefore we get an isomorphism
$$
(U+W)/U\cong W/\ker\alpha=W/(U\cap W)
$$
by the first isomorphism theorem.
I have two questions:

*

*I understand that the kernel is $U\cap W$ but how do we show that it is only $U\cap W$?


*Why does this mapping $\alpha(x)=x+U$ cover the whole space of $(U+W)/U$? (The question mentioned about coset but i thought coset only applied to multiplication not addition of x+U here)
 A: Hints:
For your first question, a double containment proof should work. That is:
Can you show $U \cap W \subseteq \text{Ker}(\alpha)$ (this shouldn't be hard, and from the sounds of your question, you may have already done it.)
Next is $\text{Ker}(\alpha) \subseteq U \cap W$. Since $\text{Ker}(\alpha) \subseteq W$ already (after all, $\alpha$ is defined on $W$), it suffices to show that $\text{Ker}(\alpha) \subseteq U$. I suggest taking some $v \not \in U$ and showing that $\alpha(v) \neq 0 + U$.

For your second question, let $v + U \in (U+W)/U$. That is, $v \in U+W$, and we are taking its $U$-coset.
We know what vectors $v \in U+W$ look like, though: $v = v_w + v_u$ with $v_w \in W$ and $v_u \in U$. Do you see how to use this to show that $\alpha$ is surjective?

I hope this helps ^_^
A: *

*If $w\in\ker\alpha(\subset W)$, $\alpha(w)=0+U$, with means that $w\in U$. But then $w\in W\cap U$.

*If $x\in(U+W)/U$, then $x=v+U$ for some $v\in U+W$. So, $v=u+w$, for some $U\in U$ and some $w\in W$. Since $u\in U$, $v+U=(u+w)+U=w+U$. But then $x=\alpha(w)$.

