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My text says:

$$ U + W = \{(x,y,0):x,y ∈ F\}$$

if

$$U = \{(x,0,0) \in F^{3} : x ∈ F\}$$ $$W = \{(0,y,0) \in F^{3} : y ∈ F\}$$

I have to verify this.

What I did was add $(x,0,0)$ and$ (y,0,0)$ together: $(x+0,0+y,0+0)$ and got $(x,y,0)$. Is that enough or there is more to it?

M.H
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Grigor
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  • :notice definition of Direct Sums is $U + W = {(x+y:x ∈ U,y\in W}$ and you can easily conclude$ U + W = {(x,y,0):x,y ∈ F}$ – M.H Mar 17 '13 at 03:07
  • so just showing the addition is enough to prove it? – Grigor Mar 17 '13 at 03:08
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    and you must showing $\dim U+\dim W=\dim(U+W)$ or $U\cap W={0}$ – M.H Mar 17 '13 at 03:13
  • @Grigor OK, I think I understand what you mean. You of course should verify $U\cap W={0}$ (or verify every element of $U+W$ has a unique expression of the form $u+w,u\in U, w\in W$ )so that the sum is a direct sum. – wxu Mar 17 '13 at 03:14
  • @wxu: The question doesn't say anything about direct sums. The title does, though, which is kind of confusing... – tomasz Mar 17 '13 at 03:17
  • it belongs to direct sum section.. the method is direct sum i guess – Grigor Mar 17 '13 at 03:25
  • linear algebra, kinda, doesn't make much sense to me.. it's a big picture which I see in parts. Can anyone point me to a good book to read, that actually makes sense and shows how everything is related? – Grigor Mar 17 '13 at 03:26
  • @Grigor: this question has been asked already, for example here: http://math.stackexchange.com/questions/143904/linear-algebra-text or here: http://math.stackexchange.com/questions/4335/where-to-start-learning-linear-algebra . – tomasz Mar 17 '13 at 03:28
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    that's why I didn't ask the question as a new question... just a comment on the side to be pointed to the right direction – Grigor Mar 17 '13 at 03:30

1 Answers1

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If it's an exercise, it's a very basic one, so I would guess you are expected to do it carefully, that is, take any element of $\{(x,y,0):x,y ∈ F\}$, and show that it belongs in $U+W$ and vice versa, explicitly writing out what you are doing and why.

If the text just says that you should verify something, then I would say that it's enough to just convince yourself that it is true and that you know how to prove it, so that would probably be enough.

tomasz
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