question.md

Consider the problem of separating $N$ data points into positive and negative examples using a linear separator. Clearly, this can always be done for $N{{,=,}}2$ points on a line of dimension $d{{,=,}}1$, regardless of how the points are labeled or where they are located (unless the points are in the same place).

1. Show that it can always be done for $N{{,=,}}3$ points on a plane of dimension $d{{,=,}}2$, unless they are collinear.
   

2. Show that it cannot always be done for $N{{,=,}}4$ points on a plane of dimension $d{{,=,}}2$.
   

3. Show that it can always be done for $N{{,=,}}4$ points in a space of dimension $d{{,=,}}3$, unless they are coplanar.
   

4. Show that it cannot always be done for $N{{,=,}}5$ points in a space of dimension $d{{,=,}}3$.
   

5. The ambitious student may wish to prove that $N$ points in general position (but not $N+1$) are linearly separable in a space of dimension $N-1$.