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

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 .

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: $$F_2\cap F_3=\{(0)\}$$
• 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} ? Feb 10, 2018 at 12:49
• 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? Feb 10, 2018 at 15:36