Hi, All:(adsbygoogle = window.adsbygoogle || []).push({});

This is a post from another site that was interesting but was not

answered:

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.

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

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