TITLE: Trouble understanding Sum of Subspaces
QUESTION [16 upvotes]: I started reviewing linear algebra, from a different textbook (Axler's), after taking a fast paced summer class. Unfortunately, I've become confused with a concept that is introduced at the end of chapter one. That is, sum of subspaces.
Axler's text defines the sum of subspaces as follows. 
Let $U_1,U_2,...,U_m$ be subspaces of a vectorspace $V$. Then we say $U_1+U_2+...+U_m=\{u_1+...+u_m:u_1\in U_1,...,u_m\in U_m\}$
I thought I understood this concept, but I'm afraid I don't because I am having trouble answering the following assertions he asks us to verify.
First that if we let $U_1,U_2,...,U_m$ be subspaces of a vectorspace $V$, then the sum of those subspaces is a subspace of $V$.
Also, this is what really had me tricked to thinking I understood it. Let $U=\{(x,0,0)\in \mathbb R^3: x\in \mathbb R\}$ and $W=\{(0,y,0)\in \mathbb R^3:y\in \mathbb R\}$, then $U+W= \{(x,y,0):x,y\in \mathbb R\}$. So this example made me think it was pretty straight forward and that I understood it, but in the next few lines he says let $Z= \{(y,y,0)\in \mathbb R^3:y\in\mathbb R\}$. Then $U+W=U+Z$ (which I am asked to verify). 
Could someone please help me understand the definition and the verifications? 
EDIT: I currently see the definition to say that when we take a collection of sets that are subspaces the sum of the sets is a set which consists of the sum  of all their elements. However when I say that it seems to me the summed set consists of just one elements (the total sum of all the elements). 
Thank You

REPLY [2 votes]: I'm also reading from the same book, this is what I understood so far.
Let's say you have an element $u \in U$, this means that it has the form of $(x, 0, 0) \in \mathbb R^3$ for some $x \in \mathbb R$. For example $(1,0,0)$ is an element in $U$ since it satisfies this definition.
Similarly, $W$ is defined as the set of all elements that satisfy $(0,y,0)\in \mathbb R^3$ for some $y\in \mathbb R$. For example $(0,\sqrt{2},0) \in W$
Now to vertify the sum $U + W$, you want to find the 'general form' an element should have in order to be in $U + W$
Let's say we add $(1,0,0)$ and $(0,\sqrt{2},0)$ to get $(1,\sqrt{2},0)$
Since this is arbitrary, we can say that the general form of the elements in the set $U + W$ will always be $(x, y, 0) \in \mathbb R^3$ for some $x, y \in \mathbb R$
Similary for $U + Z$ we can take an example: $(0.5, 0.5, 0) \in Z$ such that the sum $(1,0,0)$ and $(0.5, 0.5, 0)$ will always give me a general form of $(x, y, 0) \in \mathbb R^3$ for some $x, y \in \mathbb R$
Since the sum of two subspaces is the set of all the possible sums of elements of these subspaces, the sets $U+W$ and $U+Z$ must contain the same elements, therefore equal.