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    • Thread Starter

    I'm kinda stuck on this question:

    Let S_1,S_2 be subspaces of \mathbb{R}^n and let S_1\not={0},S_2\not={0}

    (i) Is S_1\cap S_2 a subspace?

    (ii) Prove that  S_1\cup S_2 is a subspace if, and only if  S_1\subset S_2 or S_2\subset S_1

    I've managed to do part (i), and prove the "only if" part of (ii), although I'm struggling on the "if" part of part (ii). I'm given the hint:

    if S_1\not \subset S_2, S_2\not \subset S1 consider any vectors  e_1\in S_1\setminus S_2,e_2\in S_2\setminus S_1, and prove that e_1+e_2\not \in S_1,e_1+e_2\not \in S_2, hence e_1+e_2\not \in S_1\cup S_2.

    My problem is proving that the sum doesn't lie in S_1 or S_2. Could anyone point me in the right direction?
    • PS Helper

    PS Helper
    If e_1 + e_2 \in S_1 then since e_1 \in S_1 we must have (e_1 + e_2) - e_1 \in S_1 (as it's a subspace), and then... [do the same with S_2 and derive a contradiction from here].
    • Thread Starter

    cheers ...i had a feeling thered be some sort of contradiction, just didn't know how to go about it.

    Out of curiosity, why does this require S_1 \not= {0}, S_2 \not= {0}?

    If one of them was ={0}, wouldn't that just imply that S_1 \cap S_2 = {0} which is a subspace, and S_1 \cup S_2 = whichever one isn't equal to 0 (or 0 if they're both 0) and thus also a subspace?
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