1. Assume that the network has one hidden layer. For a given assignment to the weights $\textbf{w}$, write down equations for the value of the units in the output layer as a function of $\textbf{w}$ and the input layer $\textbf{x}$, without any explicit mention of the output of the hidden layer. Show that there is a network with no hidden units that computes the same function.

2. Repeat the calculation in part (a), but this time do it for a network with any number of hidden layers.

3. Suppose a network with one hidden layer and linear activation functions has $n$ input and output nodes and $h$ hidden nodes. What effect does the transformation in part (a) to a network with no hidden layers have on the total number of weights? Discuss in particular the case $h \ll n$.

Suppose you had a neural network with linear
activation functions. That is, for each unit the output is some constant
$c$ times the weighted sum of the inputs.

1. Assume that the network has one hidden layer. For a given assignment
to the weights $\textbf{w}$, write down equations for the value of the
units in the output layer as a function of $\textbf{w}$ and the input layer
$\textbf{x}$, without any explicit mention of the output of the
hidden layer. Show that there is a network with no hidden units that
computes the same function.

2. Repeat the calculation in part (a), but this time do it for a
network with any number of hidden layers.

3. Suppose a network with one hidden layer and linear activation
functions has $n$ input and output nodes and $h$ hidden nodes. What
effect does the transformation in part (a) to a network with no
hidden layers have on the total number of weights? Discuss in
particular the case $h \ll n$.