Help understanding p-vector language Help interpreting this excerpt (annotations (numbers in parentheses) and bullets added) from The Elements of Statistical Learning

Matrices are represented by bold uppercase letters; for example, a set of $N$ input $p$-vectors (1) $x_i$, $i = 1, \ldots, N$ would be represented by the $N×p$ (2) matrix $X$. In general, vectors will not be bold, except when they have $N$ components; this convention distinguishes  (3)
  
  
*
  
*a $p$-vector of inputs $x_i$ for the $i$th observation from...
  
*...the $N$-vector $\mathbf{x}_j$ consisting of all the observations on variable $X_j$. 
  
  
  Since all vectors are assumed to be column vectors (4), the $i$th row of $X$ is $x^T_i$, the vector transpose of $x_i$ (5).

My questions: 


*

*Is there significance/convention to "$p$" in "$p$-vector"?

*Is there significance/convention to "$N$" in "$N\times p$" matrix?

*I don't understand the distinction they are making...

*What is a "column vector"?

*In the $x_i^T$ part what operation is performed first; the "$T$" (transpose) or the "$i$" (accessing an element)?


My attempts at answers:


*

*"$p$-vector" means "vector of length $p$". Since input variables ($X$) can be vectors, "$p$-vector" used in this text to refer to that case; so "$p$" represents number of elements in an input vector.

*"$N$" is used in this text to refer to when input variables ($X$) is a matrix, specifically "$N$" is used to refer to the number of rows in such a matrix. Each "row" has $p$ elements. Hence we have an $N\times p$ matrix.

*Do I illustrate the difference here:  
 

*From Wikipedia

In linear algebra, a column vector or column matrix is an $m × 1$ matrix, that is, a matrix consisting of a single column of $m$ elements,


*The language makes it clear "the vector transpose of $x_i$" means that the subscript is performed first, and then the vector transpose. Here, the transposing is probably used as described in the Wikipedia article above:


To simplify writing column vectors in-line with other text, sometimes they are written as row vectors with the transpose operation applied to them. 


This question addresses the same text as my question here, but since my question is a multipart question, and even on the subject of the "transpose" my question asks about order-of-operations, then I think my question is not completely duplicative. However, I will let moderators be the judge of that.
 A: 1 -2. Your reasoning is correct for 1 and 2, although this should be quite obvious from definition of matrix, and I am not sure what you are asking. 
3-4. Your drawing I think indicates where you are getting confused. In math, vectors are typically "column" vectors that is if $\mathbf{x}$ is a vector (or $p$-vector) I mean
$$\mathbf{x} = \begin{pmatrix} x_1 \\ \vdots \\ x_p \end{pmatrix}$$
which is contrary to your illustration. The author goes through this definition because to highlight a row he uses the following notation. 
$$
\mathbf{X} = \begin{pmatrix}
x_{11} & x_{12} & \dots & x_{1p}\\
x_{21} & x_{22} & \dots & x_{2p}\\
\vdots & \vdots & \ddots & \vdots\\
x_{N1} & x_{N2} & \dots & x_{Np}
\end{pmatrix} \qquad x_1 = \begin{pmatrix} x_{11} \\ x_{12} \\ \vdots \\ x_{1p}\end{pmatrix} \qquad x_1^\intercal = \begin{pmatrix} x_{11}& x_{12}& \dots & x_{1p}\end{pmatrix} \qquad \mathbf{x}_2 = \begin{pmatrix} x_{12} \\ x_{22} \\ \vdots \\ x_{N2}\end{pmatrix}
$$


*The author defines $x_i$ to be the column vector composed of the $p$ elements in the $i$th row of matrix $\mathbf{X}$. $x_i^\intercal$ then is simply the transpose of the vector (see above). 

