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 $(x_1, x_2, x_1^2, x_2^2, x_1 x_2)$.

FigureĀ 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
$(x_1, x_2, x_1^2, x_2^2, x_1 x_2)$.