Thread: 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!

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.

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?

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.