Define a relation on $\mathbb R^2 \setminus (0, 0)$ by letting $(x_1, y_1) \sim (x_2, y_2)$ mean that there exists a nonzero real number $\lambda$ such that $(x_1, y_1) = (λx_2, λy_2)$.
Prove that $\sim$ defines an equivalence relation on $\mathbb R^2 \setminus (0, 0)$. What are the corresponding equivalence classes and show its geometric interpretation?
Also we define the operation $[(x_1,y_1)]*[(x_2,y_2)]=[(x_1x_2-y_1y_2, x_1y_2+x_2y_1)]$, prove this is a valid definition.
I was able to prove it is an equivalence relation, however I don't know the equivalence classes (shouldn't it just be a line that goes through the centre $(0,0)$? Also how would you geometrically present this line? Or maybe i have a hole in my knowledge. I'm completely lost on the 3rd question.
Your understanding is correct: two points are equivalent if and only if they lie along the same line. To show that the multiplication is "valid" you need to show that it is well-defined, i.e. that the result does not depend on your choice of representative of the equivalence class. Namely, if $(x_1, y_1)= \lambda_1 (x_2,y_2)$ and $(x_3,y_3)=\lambda_2 (x_4,y_4)$, (i.e. take different representatives of the each equivalence class), you need to check that $$[x_1,y_1]*[x_3,y_3]=[x_2,y_2]*[x_4,y_4]$$ so that the product does not depend on your choice of representative for each equivalence class. For this, you will need to check that two sides of the equation are scalar multiples of each other, which will mean they represent the same equivalence class.