Problem $1$: Neural Network without Activation Functions $(25$ points$)$

Activation functions are crucial for neural networks to learn complex, non$-$linear relationships in the data. In this problem, consider a neural network with the following topology: an input layer with two features $(x_1$ and $x_2),$ a hidden layer with $3$ neurons, and an output layer with a single neuron. The values for weights and biases in each layer are provided below:

First layer $(l=1)$:$w_{11}^{[1]}=2,w_{12}^{[1]}=-1,w_{21}^{[1]}=1,w_{22}^{[1]}=3,w_{31}^{[1]}=-2,w_{32}^{[1]}=1,b_1^{[1]}=1,b_2^{[1]}=0,b_3^{[1]}=-1$

Second layer $(l=2)$:$w_{11}^{[2]}=1,w_{21}^{[2]}=-1,w_{31}^{[2]}=2,b_1^{[2]}=2$

Assuming there are no activation functions applied to any of the neurons, please demonstrate that the output of the network, hat$(y),$ is a linear function of the input features $x_1$ and $x_2.$

Instructions:

Use the provided weights and biases to perform forward propagation through the network.

Show all intermediate steps, including the calculations for the hidden layer and the output layer.

Conclude by proving mathematically that the final output hat$(y)$ is a linear function of $x_1$ and $x_2.$