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Question about vector space intersection properties

  1. Aug 5, 2013 #1
    Its been a while since I've done this stuff, and I don't have a text handy. I know that for sets, intersection distributes over union, I dont remember if the same will hold for vector spaces over addition?

    for example does A [itex]\cap[/itex] (B + C) = A [itex]\cap[/itex] B + A [itex]\cap[/itex] C
     
  2. jcsd
  3. Aug 5, 2013 #2

    jbunniii

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    Suppose ##A,B,C## are the following subspaces of ##\mathbb{R}^2##: ##B = \{(b,0) : b \in \mathbb{R}\}##, ##C = \{(0,c) : c \in \mathbb{R}\}##, and ##A = \{(a,a) : a \in \mathbb{R}\}##.

    Then ##B + C = \mathbb{R}^2##, so ##A \cap (B + C) = A##.

    But ##A \cap B = \{0\}## and ##A \cap C = \{0\}##, so ##(A \cap B) + (A \cap C) = \{0\}##.
     
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