Prove that two vector sub spaces intersection is ${0}$

Question

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 .

Answer 1

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?

Answer 2

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: $$F_2\cap F_3=\{(0)\}$$