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From what I’ve seen, seems like the separation hyperplane must be in the form
x.w + b = 0.
I don’t get very well this notation. From what I understand, x.w is a inner product, so it’s result will be a scalar. How can be it that you can represent a hyperplane by a scalar + b? I’m quite confused with this.
Also, even if it was x + b = 0, wouldn’t it be of a hyperplane that passes right through the origin? From what I understand a separating hyperplane doesn’t always pass through the origin!
It is the equation of a (hyper)plane using a point and normal vector.
Think of the plane as the set of points P such that the vector passing from P0 to P is perpendicular to the normal
Check out these pages for explanation:
http://mathworld.wolfram.com/Plane.html
http://en.wikipedia.org/wiki/Plane_%28geometry%29#Definition_with_a_point_and_a_normal_vector