Decision Boundary Line (Linear/Non-Linear)

[brojesus111]

Homework Statement

Given a non-linear decision boundary line: (1 + X1)^2 + (2 − X2)^2 = 4

Argue that while the decision boundary is not linear in terms of X1 and X2, it is linear in terms of X1,X1^2 , X2, and X2^2 .

The Attempt at a Solution

I'm honestly not sure. I realize the curve is a circle, but I don't understand how it could be turned linear by having it terms of X1,X1^2 , X2, and X2^2

 

[brojesus111]

Is it because we are extending the feature space by including quadratic terms that can address this non-linearity?

 

[HallsofIvy]

It's pretty basic algebra that (1+ X1)^2+ (2- X2)^2= X1^2- 2X1+ 1+ X2^2- 4X2+ 4= 4
so X1^2- 2X1+ X2^2- 4X2+ 1= 0.

If you let Y1= X1^2 and Y2= X2^2, then you have Y1- 2X1+ Y2- 4Y1+ 1= 0 which is 'linear in X1, X2, Y1, and Y2".