What is the dimension of the projection of a set of points onto a plane $Ax+By+Cz+D=0$?

Question

I know that $Ax+By+Cz+D=0$ is a plane a 2 dimensional subset of a 3D space, as it has 2 degrees of freedom (choosing the value of 2 variables determines the value of the third).

Now given the projection of a set of points onto this plane, can it have dimension 3?

Answer 1

The image is at most 2D.

When you have a point $(a,b,c)\in\mathbb{R}^3$, and you project it to a point $(a',b',c')\in\{(x,y,z)\in\mathbb{R}^3:Ax+By+Cz+D=0\}$, this point still has at most degree of freedom 2, because it still satisfies the equation of the plane $Ax+By+Cz+D=0$: setting 2 values of this equation automatically determines the value of the third one.