1 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.

a. Assume that the network has one hidden layer. For a given assignment to the weights w, write down equations for the value of the units in the output layer as a function of w and the input layer 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.

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

c. 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 n.