If $\sim$ on $\mathbb{R}^3$ is defined by:
$(x,y,z) \sim (x',y',z') \Leftrightarrow 3x - y +2z = 3x' -y' +2z'$
How are the equivalence classes described?
According to the definition:
$[(x,y,z)] = \{ (a,b,c) \in \mathbb{R}^3 | (x,y,z) \sim (a,b,c) \}$
So, for instance, if $3a-b+2c=0 \Rightarrow b = 3a+2c$, which is a ("linear") plane. The "left hand side" may take on any value in $\mathbb{R}$, not just $0$, but must equal some $3x-y+2z$ which is also a plane. So we have two planes that have the same equation, is it then correct to assume that the equivalence classes is the plane $3x-y+2z=k$, for some random value of $k$? Since the equivalence classes should be the intersection between two planes, but the planes are identical.
I'm not certain. Perhaps somebody could share more light on this.