Do neurons of a neural network model a linear relationship? I'm certain that this is a very naive question, but I am just beginning to look more deeply at neural networks, having only used decision tree approaches in the past.  Also, my formal mathematics training is more than 30 years in the past, so please be kind. :)

As I'm reading François Chollet's book on Deep Learning with Python, I'm struck that it appears that we are effectively treating the weights (kernel and biases) as terms in the standard linear equation ($y=mx+b$). At page 72 of the book, the author writes

output = dot(W, input) + b
output = (output &lt; 0 ? 0 : output)


Am I reading too much into this, or is this correct (and so fundamental I shouldn't be asking about it)?
In a neural network (NN), a neuron can act as a linear operator, but it usually acts as a non-linear one. The usual equation of a neuron $i$ in layer $l$ of an NN is 

$$o_i^l = \sigma(\mathbf{x}_i^l \cdot  \mathbf{w}_i^l + b_i^l),$$ 

where $\sigma$ is a so-called activation function, which is usually a non-linearity, but it can also be the identity function, $\mathbf{x}_i^l$ and $\mathbf{w}_i^l$ are the vectors that respectively contain the inputs and the weights for neuron $i$ in layer $l$, and $b_i^l \in \mathbb{R}$ is a bias. Similarly, the output of a layer of a feed-forward neural network (FFNN) is computed as

$$\mathbf{o}^l = \sigma(\mathbf{X}^l \mathbf{W}^l + \mathbf{b}^l).$$

In your specific example, you set the new weight to $0$, if the output of the linear combination is less than $0$, else you use the output of the linear combination. This is the definition of the ReLU activation function, which is a non-linear function.