While studying the basics neccessary for deep learning, I read that for any matrix $\textbf{W}$ and vectors $\textbf{w},\textbf{x}$ with appropriate dimensions so the multiplications are defined, it holds that:
$\textbf{w}^T \textbf{W}^T \textbf{x} = \textbf{x}^T \textbf{W} \textbf{w}$
However I am not sure how to arrive at this. I know about the general rule for transposing matrix products which states that the transpose of a product is the product of the transposes in reverse order:
$(\textbf{A}\textbf{B})^T=\textbf{B}^T\textbf{A}^T$
So would the derivation be that $\textbf{w}^T \textbf{W}^T \textbf{x} = (\textbf{x}^T \textbf{W} \textbf{w})^T$ and since the result of $\textbf{x}^T \textbf{W} \textbf{w}$ is a scalar we can say $(\textbf{x}^T \textbf{W} \textbf{w})^T = \textbf{x}^T \textbf{W} \textbf{w}$ since the transpose of a scalar is the scalar itself?
I am no Mathematician so this question might seem trivial for many of you, however any feedback is highly appreciated.
Yes, you can do the simple calculation: Let w a matrix Nx1; Let W a matrix MxN; Let x a matrix 1xM;
After calculate the product, you'll see the result is a scalar "1x1", and so you're right.