Syntax Explanation - Tranpose of Dot Product

Question

On page 29 of https://arxiv.org/pdf/1802.01528.pdf there is syntax that I find confusing:

What does the syntax [wT, b]T mean? Is this a 2 dimensional matrix formed from a horizontal row w (transposed from column) and a column of b, all transposed? What is the final shape? Same question for x hat.

Last, how does w.x+b then become w hat . x hat?

Answer 1

If $\mathbf{w} = \begin{bmatrix}w_1 \\ w_2 \\ \vdots \\ w_n\end{bmatrix}$ then $$\hat{\mathbf{w}} = \begin{bmatrix}w_1 \\ w_2 \\ \vdots \\ w_n \\ b \end{bmatrix}.$$ Similarly $$\hat{\mathbf{x}} = \begin{bmatrix}x_1 \\ \vdots \\ x_n \\ 1\end{bmatrix}.$$

Finally, $$\hat{\mathbf{w}} \cdot \hat{\mathbf{x}} = w_1 x_1 + w_2 x_2 + \cdots + w_n x_n + b =\mathbf{w} \cdot \mathbf{x} + b.$$

Answer 2

If we have $$w = \begin{bmatrix} w_{1}\\w_{2}\\\vdots\\w_{n}\end{bmatrix}$$ then $w^{T} = [w_{1}, \ldots, w_{n}]$, $[w^{T},b] = [w_{1}, \ldots, w_n, b]$, and $$[w^{T},b] = [w_{1}, \ldots, w_n, b]^{T} =\begin{bmatrix}w_{1}\\\vdots\\w_{n}\\b \end{bmatrix}.$$