I am self studying some math subjects to help me with my material science studies, as I enjoy mathematical rigor more than using ready made equations/tools. I mention this to note that unfortunately right now, my math background is very thin besides some calculus, integrals etc. I was reading "Linear algebra done right" by Axler and I am a bit confused on these two specific examples for the sums of subsets.
Example 1.37
Suppose $U$ is the set of all elements of $F^3$ whose second and third coordinates equal $0$, and $W$ is the set of all elements of $F^3$ whose first and third coordinates equal $0$:
$$U = \{(x,0,0) \in F^3 : x \in F\}$$ and $$W = \{(0,y,0) \in F^3 : y \in F\}$$
Then
$$U + W = \{(x,y,0) : x,y \in F\}$$
as you should verify.
Example 1.38
Suppose that $U = \{(x,x,y,y) \in F^4 : x,y \in F\}$ and $W = \{(x,x,x,y) \in F^4 : x,y \in F\}$. Then
$$U + W = \{(x,x,y,z) \in F^4: x,y,z \in F\}$$
as you should verify.
I understand the logic in the second example, that the variable names are local and can just be replaced with a,b,c to avoid confusion. This question was about that and the answers made it clear. But I have trouble understanding what the actual question is, what am I supposed to prove/verify? I posted the first example too, because it is very simple and I would like to understand that specific case too.
In a more general problem, suppose I have two sets $U$ and $W$ and I want to find their sum, what is the formal procedure I have to follow? Can I just add their elements and be done with it or do I have to prove the set equality after (one is the subset of the other and vice versa)? I guess that seems trivial for the two examples I posted because of the closed under addition property of the field $F$ but I would think that more complex problems would not be as trivial.
What I tried in example 1.37:
The set to verify as the result of addition is
$$S = \{(x,y,0) : x,y \in F\}$$
and
$u = (x,0,0) \in U$, $w = (0,y,0) \in W$, $u + w = (x+0,0+y,0+0)=(x,y,0)$, therefore $u+w \in S$ ($x,y \in F$).
So, we proved that $U+W \subseteq S$, but how do we prove that $S \subseteq U + W$? I have no idea how to start. In the answer to this question for example 1.38, he "fixes" an element $s$ of $S$ as the result we were given, $(x,x,y,z)$, and then chooses $u$ and $w$ so that they fit. Is intuitively setting elements values what you are supposed to do in all similar problems or is there a more formal methodology? And do I have to do this for every sum of sets/subsets I want to find?
I understand this is a lot to unpack, but just some nudge to a general direction would help me a lot. I would just comment on the answer but I don't have the required reputation.
Best Answer
If $(x,y,0)\in S$, then$$(x,y,0)=\overbrace{(x,0,0)}^{\phantom U\in U}+\overbrace{(0,y,0)}^{\phantom V\in V}.$$
Concerning the other problem, since the first coordinate of every element of $U$ is equal to the second one and since the same thing occurs with $V$, it is clear that, if$$T=\{(x,x,y,z)\mid x,y,z\in F\},$$then $U+V\subset T$. Now, take $(x,x,y,z)\in T$ and consider the equation $$(x,x,y,z)=(a,a,b,b)+(c,c,c,d),$$which is equivalent to$$\left\{\begin{array}{l}x=a+c\\y=b+c\\z=b+d.\end{array}\right.$$This system has solutions; for instance, you can take $a=0$, $b=-x+y$, $c=x$, and $d=x-y+z$. This proves that every element of $T$ can be written as an element of $U$ plus an element of $V$. Therefore, $T\subset U+V$.