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

[WWGD]

Hi, All:
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.

 

[Ben Niehoff]

Sounds correct to me.