Confusion with Notation in the Book on Deep Learning by Ian Goodfellow et al In chapter 6.1 on 'Example: Learning XOR', the bottom of page 168 mentions:

The activation function $g$ is typically chosen to be a function that
is applied element-wise, with $h_i = g(x^TW_{:,i}+c_i).$

Then we see equation 6.3 is defined as (assuming g as ReLU):

We can now specify our complete network as
$f(x; W,c,w,b) = w^T$
max$\{0, W^Tx + c\} + b$

Wondering why the book uses $W^Tx$ in equation 6.3, while I expect it to be $x^TW$. Unlike XOR example in the book where $W$ is a $2\times2$ square matrix, we may have non-square $W$ as well, and in such cases, $x^TW$ is not same as $W^Tx$.
Please help me understand, if I'm missing something here.
Let $\mathbf{y}  = \mathbf{W}^T \mathbf{x}$
Then, $\mathbf{y}^T =(\mathbf{W}^T \mathbf{x})^T =\mathbf{x}^{T}(W^T)^T = \mathbf{x}^{T}W $. Note that $\mathbf{W}$ does not have to be a square matrix.
Let $e^{(i)}_{j} = \delta_{i,j} $.
Then,  $y_{i} = \mathbf{y}^{T}e^{(i)} = (\mathbf{x}^T W) e^{(i)} = \mathbf{x}^{T}(We^{(i)}) = \mathbf{x}^{T}W_{:,i}$
and thus
$h_{i} = g(\mathbf{x}^T W_{:,i}+c_{i}) = g(y_{i}+c_{i})$
On the other hand,
$f(..)  = w^{T} \max\{\mathbf{0},W^{T}\mathbf{x}+\mathbf{c}\}+b = w^{T} \max\{\mathbf{0},\mathbf{y}+\mathbf{c}\}+\mathbf{b}$.
Does that answer your question ?