In the weight update rule of the Perception Learning Algorithm $($referring to week $1\text{'}$s slide page $64),$

we observe that the weights moves in the direction of classifying $x_{**}$ correctly.

The rule can be written as: $w(t+1)=w(t)+y_{**}{x}_{**}.$

Please note that we have: $x=\{1\}{R}^d=\{[x_0,x_1,$cdots,$x_d{]}^T|x_0=1,x_1$inR,cdots,$x_d$inR$\}$

$(1)$ Prove that $y_{**}{w}^T(t)x_{**}<0.$

Hint: $x_{**}$ is misclassified by $w(t).$

$(2)$ Prove that $y_{**}{w}^T(t+1)x_{**}>y_{**}{w}^T(t)x_{**}.$

Hint: Use the weight update rule.