1. Suppose that a list contains the values 20 44 48 55 62 66 74 88 93 99 at index positions 0 through 9. Trace the values of the variables….

## Show how to express h(x) on the basis of the kernel function, and without accessing individual entries of ψ(x) or w.

1. Let w = *c*+ −*c*− and let *b *= 12 ( *c*− 2 − *c*+ 2). Show that *h*(x) = sign(_w*,ψ*(x)_+*b*).

2. Show how to express *h*(x) on the basis of the kernel function, and without accessing individual entries of *ψ*(x) or w.

3. Consider a set *S *of examples in R*n *× [*k*] for which there exist vectors *μ*1*, . . .,**μ**k *such that every example (x*, **y*) ∈ *S *falls within a ball centered at *μy *whose radius* *is *r *≥ 1. Assume also that for every *i *_= *j *, *μ**i *−*μ**j* ≥ 4*r *. Consider concatenating* *each instance by the constant 1 and then applying the multivector construction,* *namely,* **_*(x*, **y*)= [ :0*, .*;*.**. ,*0= ∈R(*y*−1)(*n*+1)* **, *:*x*1*, . .*;*.**, **xn **,*1=* *∈R*n*+1 *, *:0*, .*;*.**. ,*0=* *∈R(*k*−*y*)(*n*+1)* *].

Show that there exists a vector w ∈ R*k*(*n*+1) such that * *(w*, *(x*, **y*)) = 0 for every (x*, **y*) ∈ *S*.

*Hint: *Observe that for every example (x*, **y*) ∈ *S *we can write x = *μ**y *+v for some v ≤*r*. Now, take w = [w1*, . . .,*w*k *], where w*i *= [*μi **, *− *μ**i* 2*/*2].