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I'm having a hard time to prove that two vector sub spaces intersection is the zero space .

I know that I have to prove that they are a direct sum , but I don't really know how to prove that, all I know is the following :

  • $F_1, F_2, F_3$ are subspaces
  • $F_1 \cap F_2 = (0) $
  • $(F_1 + F_2) \cap F_3 = ( 0 )$

I need to prove that : $F_2 \cap F_3 = (0)$ and that $F_1 \cap (F_2 + F_3) = (0)$ , Please don't answer both, just explain how to do the first one and give me a hint to do the second .

Thank's .

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You know that F1 and F2 have no elements in common. Therefore you know F1 + F2 has unique elements. The intersection between (F1 + F2) and F3 is also 0. What follows is that the intersection between F1 and F3 and the intersection between F2 and F3 are also zero. Does this make sense?

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  • $\begingroup$ No I didn't really understand what you mean $\endgroup$
    – Souames
    Feb 10 '18 at 12:16
  • $\begingroup$ Chose F1: a,b,c ; F2: d,e,f ; F3: x,y,z, We know the intersection between F1 and F2 is 0 (this is true in this case). Now if we add the elements in F1 and F2, we get a,b,c,d,e,f. And we also know that (F1 + F2) and F3 have no elements in common (also true in this case). Because F1 and F2 have no elements in common, the sum doesn't have elements in common either. Therefore (F1 + F2), F1, F2, F3 have all unique elements. Now complete the proof. $\endgroup$
    – mussolo
    Feb 10 '18 at 12:23
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For the first question, we know $F_2$ is a subset of $F_1+F_2$: $$F_2\subseteq F_1+F_2$$ and therefore its intersection with $F_3$ is a subset of the intersection of $F_1+F_2$ with $F_3$, which is the single element set $\{(0)\}$: $$F_2\cap F_3\subseteq (F_1+F_2)\cap F_3=\{(0)\}$$ Since, $F_2$ and $F_3$ are subspaces, they both have the zero element, so their intersection is: $$F_2\cap F_3=\{(0)\}$$

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  • $\begingroup$ Thank's, but I'm stuck on the second, I know I have to use the new result, I've proven that $0$ is a subset of $ F_1 \cap (F_2 + F_3)$ (because they all contain the zero element since they are subspaces), in order to prove the equality I need to prove the inclusion on both sides .so how do I prove the inverse now ? i.e : $F_1 \cap (F_2 + F_3)$ is a subset of {0} ? $\endgroup$
    – Souames
    Feb 10 '18 at 12:49
  • $\begingroup$ Hint: assume that $x\in F_1\cap (F_2+F_3)$. Now, you can say $x\in F_1$ and $x\in F_2+F_3$, which means that there are $x_2\in F_2$ and $x_3\in F_3$ that satisfy $x=x_2+x_3$. Can you follow that? $\endgroup$
    – Babak
    Feb 10 '18 at 15:36

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