Adding a single hidden layer, the math gets quite verbose. It is easier to treat an input to a node as a transformation matrix \(T\). For the single hidden layer NN, there are two transformations \(T^{(1)}\) and \(T^{(2)}\).
Let’s begin with a single sample \(x \in \mathbb R^{d}\). At every step along the way we will always add the ficticious dimension of value 1 for the intercept.
\(T^{(1)}\) takes in \(x\) as \(d+1\) input nodes and outputs \(h \in \mathbb R^{m}\)
\(T^{(2)}\) takes in \(h\) as \(m+1\) nodes and outputs \(z \in \mathbb R^{k}\)