First I'd like to say that Professor Abu-Mostafa's lectures are unsurpassed in clarity and effectiveness in communicating understanding of the key elements of this fascinating subject. So it is unusual (actually, unique so far) for me to think I can see a way in which clarity of a point can be improved. Maybe I'm wrong: please judge!

The proof I am referring to is the second side of the proof of the VC dimension of a perceptron, where it is necessary to show that no set of

points can be shattered. Here's my slightly different version.

Consider a set

of

points in

-dimensional space, all of which have first co-ordinate 1. There is a non-trivial linear relation on these points:

Rearrange so all the coefficients are positive (changing labels for convenience)

and

must be non-empty subsets of

because the relation is non-trivial, and the first co-ordinate of all the points is 1.

If some perceptron is positive on

and negative on

then the value of the perceptron on

is positive and its value on

is negative.

But from the above these are the same point. Hence such a perceptron does not exist, so

cannot be shattered, completing the proof.