# Question

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 labelled 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. From this it follows that the VC dimension of linear half spaces in dimension N — 1 is N.

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. From this it follows that the VC dimension of linear half spaces in dimension N — 1 is N.

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