I am working with subspaces and using Linear Algebra done right as literature. Axler defines the sum of subsets as:
Suppose $U_1,...,U_m$ are subsets of V. The sum of $U_1,...,U_m$, denoted $U_1+\cdots+U_m$, is in the set of all possible sums of elements of $U_1,...,U_m$. More precisely, $$U_1+\cdots+U_m=\{u_1+\cdots+u_m:u_1\in U_1,\dots,u_m\in U_m\}$$
Now suppose that $U = \{(x,x,y,y)\in\mathbb{F}^4:x,y\in\mathbb{F}\}$ and $W = \{(x,x,x,y)\in\mathbb{F}^4:x,y\in\mathbb{F}\}$. Then:
$$U+W=\{(x,x,y,z)\in\mathbb{F}^4:x,y,z\in\mathbb{F}\}$$
When I try to verify the assertion I get:
$$U+W = \{(x+x,x+x,x+y,y+y):x,y\in\mathbb{F}\}$$
I just apply the definition of the sum of subsets. What am I missing? Any help greatly appreciated.
I also had trouble understanding this one, but I think it all comes down to how you name the elements of the set.
As in this answer, and this comment, we can use different letters for one of the sets. Let's suppose that $U = \{(x,x,y,y)\in\mathbb{F}^4:x,y\in\mathbb{F}\}$ and $W = \{(v,v,v,w)\in\mathbb{F}^4:v,w\in\mathbb{F}\}$.
Now, we get the same as your result:
$$U+W = \{(x+v,x+v,y+v,y+w):x,y,v,w\in\mathbb{F}\}$$
Then, by setting $x=x+v$, $y=y+v$, $z=y+w$, we have
$$U+W = \{(x, x, y, z):x,y,z\in\mathbb{F}\}$$
Which is what the book directly states.