### Artificial IntelligenceAIMA Exercises

Consider the problem of separating $N$ data points into positive and negative examples using a linear separator. Clearly, this can always be done for $N2$ points on a line of dimension $d1$, 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 $N3$ points on a plane of dimension $d2$, unless they are collinear.
2. Show that it cannot always be done for $N4$ points on a plane of dimension $d2$.
3. Show that it can always be done for $N4$ points in a space of dimension $d3$, unless they are coplanar.
4. Show that it cannot always be done for $N5$ points in a space of dimension $d3$.
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 $N-1$.

Consider the problem of separating $N$ data points into positive and negative examples using a linear separator. Clearly, this can always be done for $N2$ points on a line of dimension $d1$, 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 $N3$ points on a plane of dimension $d2$, unless they are collinear.
2. Show that it cannot always be done for $N4$ points on a plane of dimension $d2$.
3. Show that it can always be done for $N4$ points in a space of dimension $d3$, unless they are coplanar.
4. Show that it cannot always be done for $N5$ points in a space of dimension $d3$.
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 $N-1$.

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