Real vs. integer non-paradox

*To*: mathgroup at yoda.physics.unc.edu*Subject*: Real vs. integer non-paradox*From*: JWENDEL at isdmnl.wr.usgs.gov*Date*: Thu, 11 Mar 1993 15:32:07 -0800 (PST)

There's really not much of a paradox if any. The bases found for NullSpace[A] and NullSpace[Ar] look very different, to be sure, but they actually generate the same spaces. This may be seen in various ways. One is to compute the angles between them as shown in the book Matrix (pg. 581) by Van Loan & Golub; this may be done nicely by using QRDecomposition and SingularValues in Mma. Alternatively one may use Fit[{{x's,y's,f's},...},{x,y},{x,y}] [no 1's!] to see that the vectors in the basis given by NullSpace[Ar] are linear combinations of those given by NullSpace[A]. I forgot to say in the first part that the angles turn out to have cosines equal to 1.0 to the default precision employed, i.e. the angles are zero.