Question $1$: An artificial neuron with linear activation function, two inputs and a bias is

trained using the rules:

$w^{t+1}=w^t+x^t,$ if $x^t$ belongs to class$-1$ and the transpose of $w^t$ multiplied

with $x^t$ is less than or equal to zero;

$w^{t+1}=w^t-x^t,$ if $x^t$ belongs to class$-2$ and the transpose of $w^t$ multiplied

with $x^t$ is greater or equal to zero;

$w^{t+1}=w^t$ otherwise.

In the above expressions, $w$ is the weight vector, $t$ denotes iterations, and $x$ is the feature vector

as shown in the following table:

All weights are zero at the start, i$.$e$.w^0=[0,0,0{]}^T,$ where $T$ indicates the transpose of the

vector, the first zero element in this vector is the initial weight of the first input $x_1,$ the second

zero is the initial weight of the second input $x_2,$ and the last zero element in this vector is the

weight of the bias term, which always receives an input equal to $1.$ The neuron is trained by

feeding input vectors in the order presented in the table, i$.$e$.t=0,1,2,3.$ For example, starting

from the top, at $t=0,$ the first input pattern is $-1,0$ and the correct classification for this input

vector is "class$-1".$

$($Q$1\mathrm{.}1)$ What would the weights be after the presentation of each one of the input patterns?

Show and explain all your calculations.

$(10$ marks$)$

$($Q$1\mathrm{.}2)$ What would the weights be if you continue training for three more iterations, i$.$e$.$

$t=4,5,6?$ Show and explain your calculations. What is the weight vector at the end of training

$(t=6)?$ Show and explain all your calculations.

$(10$ marks$)$

$($Q$1\mathrm{.}3)$ Has the training algorithm reached convergence at $t=6?$ Explain your view. What is the

equation of the line that defines the decision boundary of this neuron at $t=6?$ Show and explain

all your calculations.