We define a relation $\sim$ on $\mathbb{R}^2$ $\mathbin{/}$ $\{0,0\}$ by chosing two points $p, q \in {\mathbb{R}²}$, if they are on the same line, where the line has to go through the origin (0,0). Show that $\sim$ is a equivalence relation and define its quotient space and the equivalence classes.
So each line that goes through the origine can be written as $y=m \cdot x$, so if we choose $m$ arbitrarily but fixed than each point which lies on this line has to fulfill $y_{p}=m \cdot x_{p} $. showing reflexivity and transitivity and symmetry is not a problem here, however I do not know what the equivalence classes and the quotient space would be?