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Dimension proof of the intersection of 3 subspaces

  1. Jan 13, 2013 #1
    1. The problem statement, all variables and given/known data

    Assume [itex]V = \mathbb{R}^n[/itex] where [itex]n \geq 3[/itex]. Suppose that [itex]U,W,X[/itex] are three distinct subspaces of dimension [itex]n-1[/itex]; is it true then that [itex]dim(U \cap W \cap X) = n-3[/itex]? Either give a proof, or find a counterexample.

    3. The attempt at a solution

    The question previous to this was showing that for subspaces [itex]U,W[/itex] of dimension [itex]n-1 \longrightarrow dim(U \cap W) = n - 2[/itex] which I was able to prove fine. Now, I'm thinking this is true, because [itex]W \cap X[/itex] will be a subspace of dimension [itex]n-2[/itex] so if we set [itex]W \cap X = Z[/itex], where [itex]Z[/itex] is a subspace we turn this problem into [itex]dim(U \cap Z)[/itex] and since all three were distinct we should have [itex]dim(U \cap Z) = n-3[/itex] but I'm not entirely sure.

  2. jcsd
  3. Jan 13, 2013 #2
    You seem to be on the right track.
  4. Jan 13, 2013 #3
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