### Artificial IntelligenceAIMA Exercises

Figure kernel-machine-figure showed how a circle at the origin can be linearly separated by mapping from the features $(x_1, x_2)$ to the two dimensions $(x_1^2, x_2^2)$. But what if the circle is not located at the origin? What if it is an ellipse, not a circle? The general equation for a circle (and hence the decision boundary) is $(x_1-a)^2 + (x_2-b)^2 - r^20$, and the general equation for an ellipse is $c(x_1-a)^2 + d(x_2-b)^2 - 1 0$.
1. Expand out the equation for the circle and show what the weights $w_i$ would be for the decision boundary in the four-dimensional feature space $(x_1, x_2, x_1^2, x_2^2)$. Explain why this means that any circle is linearly separable in this space.
2. Do the same for ellipses in the five-dimensional feature space $(x_1, x_2, x_1^2, x_2^2, x_1 x_2)$.

Figure kernel-machine-figure showed how a circle at the origin can be linearly separated by mapping from the features $(x_1, x_2)$ to the two dimensions $(x_1^2, x_2^2)$. But what if the circle is not located at the origin? What if it is an ellipse, not a circle? The general equation for a circle (and hence the decision boundary) is $(x_1-a)^2 + (x_2-b)^2 - r^20$, and the general equation for an ellipse is $c(x_1-a)^2 + d(x_2-b)^2 - 1 0$.
1. Expand out the equation for the circle and show what the weights $w_i$ would be for the decision boundary in the four-dimensional feature space $(x_1, x_2, x_1^2, x_2^2)$. Explain why this means that any circle is linearly separable in this space.
2. Do the same for ellipses in the five-dimensional feature space $(x_1, x_2, x_1^2, x_2^2, x_1 x_2)$.

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