Trouble understanding vector subspace sum

[Cyview]

I'm self-studying Linear Algebra and the book I'm using is Linear Algebra done right by Sheldon Axler but I came across something that I don't understand .-
Suppose $\mathrm U$ is the set of all elements of $\mathbb F ^3$ whose second and third coordinates equal 0, and $\mathrm W$ is the set of all elements of $\mathbb F ^3$ whose first and third coordinates equal 0:
$$\mathrm U = \{ (x , 0 ,0) \in \mathbb F^3: x \in \mathbb F \} \text{ and } \mathrm W= \{(0,y,0) \in \mathbb F^3:y \in F \}$$
then
$$\mathrm U + \mathrm W = \{ (x, y, 0) : x, y \in \mathbb F \}$$
As another example, suppose $\mathrm U$ is as above and $\mathrm W$ is the set of all elements of $\mathbb F^3$ whose first and second coordinates equal each other and whose third coordinate equals 0:
$$\mathrm W= \{(y,y,0) \in \mathbb F^3:y \in F \}$$
Then $\mathrm U + \mathrm W$ is also given by.- (this is the part I don't understand)
$$\mathrm U + \mathrm W = \{ (x, y, 0) : x, y \in \mathbb F \}$$

why is that it's the same result nevertheless the subspace has changed?
it shouldn't be something like $$\mathrm U + \mathrm W = \{ (x + y, y, 0) : x, y \in \mathbb F \}$$ what am I missing here?, thank you very much.

 

[vela]

The $x$ in the definition of U and the $x$ in the definition of U+W aren't the same. In both cases, U+W is the same subspace of F³, i.e. vectors of the form (x,y,0).

 

[Dick]

$$\mathrm U + \mathrm W = \{ (x + y, y, 0) : x, y \in \mathbb F \}$$ is the same thing as $$\mathrm U + \mathrm W = \{ (u, v, 0) : u, v \in \mathbb F \}$$ where u=x+y and v=y. Given any u and v you can solve for x and y and vice versa. So they are the same subspace.

 

[Cyview]

Thank you very much, now I understand that it just serve as an arbitrary variable to denote all the possible values, it's the first time I read a book as rigorous as this one, but know I get it