Help understanding p-vector language

Question

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)

1. a $p$-vector of inputs $x_i$ for the $i$th observation from...
2. ...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:

1. Is there significance/convention to "$p$" in "$p$-vector"?
2. Is there significance/convention to "$N$" in "$N\times p$" matrix?
3. I don't understand the distinction they are making...
4. What is a "column vector"?
5. In the $x_i^T$ part what operation is performed first; the "$T$" (transpose) or the "$i$" (accessing an element)?

My attempts at answers:

1. "$p$-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.
2. "$N$" 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.

3. Do I illustrate the difference here:

   

4. 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,

5. 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.

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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.

Answer 1

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.

$$ \mathbf{X} = \begin{pmatrix} x_{11} & x_{12} & \dots & x_{1p}\\ x_{21} & x_{22} & \dots & x_{2p}\\ \vdots & \vdots & \ddots & \vdots\\ x_{N1} & x_{N2} & \dots & x_{Np} \end{pmatrix} \qquad x_1 = \begin{pmatrix} x_{11} \\ x_{12} \\ \vdots \\ x_{1p}\end{pmatrix} \qquad x_1^\intercal = \begin{pmatrix} x_{11}& x_{12}& \dots & x_{1p}\end{pmatrix} \qquad \mathbf{x}_2 = \begin{pmatrix} x_{12} \\ x_{22} \\ \vdots \\ x_{N2}\end{pmatrix} $$

5. 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).