1 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).

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

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

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

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

e. 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.