Exercise 1.3

From the previous problem, we see that $y(t)w^T(t)x(t)$ is increasing with each update.

If $y(t)$ is positive, but $w^T(t)x(t)$ is negative, we move $w^T(t)x(t)$ toward positive by increasing it.

If however $y(t)$ is negative, but $w^T(t)x(t)$ is positive, $y(t)w^T(t)x(t)$ increases means $w^T(t)x(t)$ is decreasing, i.e. moving toward negative region.

So the move from $w(t)$ to $w(t+1)$ is a move “in the right direction” as far as classifying $x(t)$ is concerned.