Why is the sum U + W = {(x, y, 0)} not {(x+y, y, 0)}?

[airpocket]

The sum of two subspaces seems a simple enough concept to me, but I must be misunderstanding it since I don't understand why Axler gives an answer he does in Linear Algebra Done Right.

Suppose U and W are subspaces of some vector space V.

$$U = \{(x, 0, 0) \in \textbf{F}^3 : x \in \textbf{F}\} \text{ and } W = \{(y, y, 0) \in \textbf{F}^3 : y \in \textbf{F}\}.$$

The sum is given as follows:

$$U + W = \{(x, y, 0) : x, y \in \textbf{F}\}.$$

Whereas it seems to me it should be:

$$U + W = \{(x+y, y, 0) : x, y \in \textbf{F}\}.$$

Am I misunderstanding the concept of sum, and it does not really mean that all the elements in $$U + W$$ should have the form $$(x, 0, 0) + (y, y, 0)$$, or $$(x+y, y, 0)$$?

 

[mathwonk]

since x can be anything, it can be anything -y, so the set of vectors of form (x,y) where x and y bare anything, is the same as the set of vectors (x+y,y) where x and y are anything.

so you are also right!

 

[airpocket]

Thanks so much. That makes sense. I didn't think about simplifying x + y.