# Equivalence classes for an particular relation question

• Apr 20th 2011, 05:29 AM
LHS
Equivalence classes for an particular relation question
Hi,

If anyone could help me with this I would be very glad! I have said that M=(aij) and M^T=M^-1
therefore if e1 relates v, where v=(x,y,z) then v=(a11,a21,a31) and all of those values can't be simultaneously zero for M^-1 to exist.. can't seem to get any further!

http://img15.imageshack.us/img15/1/unledjs.png
• Apr 20th 2011, 06:32 AM
HappyJoe
Hello.

Maybe it helps you, if you recall that orthogonal matrices always preserve the norm of a vector. So if v is some vector, then the lenght of Mv is equal to the lenght of v, when M is orthogonal.

So certainly the equivalence class of e_1 can only contain vectors of lenght 1. The question remains if the equivalence class contains all vectors of lenght 1.
• Apr 20th 2011, 06:35 AM
LHS
so would it be ok to say that as MM^T=I then a11^2+a21^2+a31^2=1 therefore [e1]={ (x,y,z) : x^2+y^2+z^2=1 } as a method of proving that is's a sphere of radius 1?
Then for the others, you can say they are concentric spheres, for for the equivalence class of (a,b,c) is it a sphere with radius (a^2+b^2+c^2)^0.5? Think that's enough justification?
• Apr 20th 2011, 03:38 PM
HallsofIvy
Notice that HappyJoe only said that all vectors in that equivalence class have length 1 (and it follows that all vectors in any given equivalence class have the same length). He then suggested that you determine whether or not all vectors of length 1 are in that equivalence class.