What does it mean by transpose of a vector

Question

I came across this paragraph (confusing transpose signs) in a matrix calculus paper. I'm confused by the double transpose notation of the vectors, in particular t(w). I thought w hat should be t[w, b] instead of t[t(w), b], and x hat should be [x, 1] instead of [t(x), 1]. Could you please confirm whether the highlighted transpose signs in the attached are correct? Thanks

Answer 1

In my opinion, the first is correct, while the second one should contain a typo.

\begin{align*} \mathbf{w}^T&=[w_1,...,w_n]\\ \hat{\mathbf{w}}^T&=[\mathbf{w}^T,b]\\ &=[w_1,...,w_n,b], \end{align*} and after applying transpose on the above row vectors, we have $ \hat{\mathbf{w}}=[\mathbf{w}^T,b]^T$, a column vector.

Similarly, \begin{align*} \mathbf{x}^T&=[x_1,...,x_n]\\ \hat{\mathbf{x}}^T&=[\mathbf{x}^T,1]\\ &=[x_1,...,x_n,1], \end{align*} and then we have $\hat{\mathbf{x}}=[\mathbf{x}^T,1]^T$, a column vector.

Answer 2

You see vectors are usually written in columns, also known as $n\times1$ matrices. Take for example the $3\times1$ matrix (or column vector) $w = \left(\begin{smallmatrix}x_1\\x_2\\x_3\end{smallmatrix}\right)$. So $w^T$ is the $1\times3$ matrix (or row vector) $(x_1, x_2, x_3)$, and if we define $\hat{w}= (w^T, b)^T$ then that is equal to $(x_1,x_2,x_3,b)^T=\left(\begin{smallmatrix}x_1\\x_2\\x_3\\b\end{smallmatrix}\right)$.