I am trying to understand the Matrix and Vector Notations on page 2 here: (the page is also pasted below, to make it easier to explain the problem).
Problem: For equation (2), I think it should be $\textbf{h}^{T}$ instead of $\textbf{h}$ for matrix multiplication to make sense:
Why I think it should be $\textbf{h}^{T}$
$\textbf{x}$ is a Vx1 dimensional vector. $\textbf{W}$ is a VxN dimensional Matrix. Below I re-write equation (1) with dimensions for each vector and matrix:
$$ \textbf{h}_{1xN} = \textbf{x}^{T}_{1xV}\textbf{W}_{VxN} = {\textbf{v}_{w_{I}}}_{1xN}$$
$\textbf{v}^{'}_{w_{j}}$ is a Nx1 dimensional j-th column of matrix $\textbf{W}^{'}$ and $u_{j}$ is a scalar. Therefore rewriting equation (2) with dimensions (you can only multiple when you use $\textbf{h}^{T}$ and not $\textbf{h}$ as shown in figure):
$${u_{j}}_{1x1} = {\textbf{v}^{'}_{w_{j}}}^{T}_{1xN}.\textbf{h}^{T}_{Nx1}$$