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Intersection of Planes in R^3 and Dense Subsets of R^3

  1. Sep 12, 2011 #1


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    Hi, All:
    This is a post from another site that was interesting but was not

    can I reasonably
    > argue that three planes in 3-space are not likely
    > to intersect at a point using the fact that
    >t GL(3,R);
    > the subset of invertible 3x3-matrices has measure 0
    > in M(n,R); the set of all 3x3-matrices?
    > Basically, the intersection of three planes Pi:=
    > a_ix+b_iy+c_iz =d_i ; i=1,2,3.
    > Is the same as having the matrix M with rows
    > (a_i b_i c_i ) can be reduced to Jordan form
    > with all 1's on the diagonal, and this is the
    > same as M being invertible.

    i.e., if M is invertible, then it can be turned into the Jordan Form as
    the identity, which means that Ax=b will have a solution, with b=(d_1,d_2,d_3)
    as above, i.e., the d_i are the constant terms.

    Seems reasonable; wonder what others think.
  2. jcsd
  3. Sep 12, 2011 #2

    Ben Niehoff

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    Sounds correct to me.
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