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

The above text is taken from chapter 2 page 10, I understand the first sentence, data is represented in a matrix where there are p variables and N observations. What I don't understand is this bit:

this convention distinguishes a p-vector of inputs $x_i$ for the $i^{th}$ observation from the N-vector $\textbf{x}_j$ consisting of all the observations on variable $X_j$

Can someone please clarify this?

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Let's start with the understanding of what $X \in R^{\text{N x p}}$ is:

It is a matrix with dimension $\text{N x p}$, which means that $X$ has $N$ rows and $p$ columns.

Now what is $N$ ? - The number of observations.

So what is $p$ ? - The number of variables or features.

Further, recall that each of the N rows corresponds to a singe observation, giving the p variables/features used in a learning algorithm. Therefore, the i-th row vector stores all values of features for individual i and is of dimension $R^\text{1 x p}$

The j-th column-vector therefore corresponds to the vaues of variable j for each of the $N$ observations and is of dimension $R^\text{N x 1}$

With the last sentence they literally say: column-vectors are denoted in bold, row-vectors not.

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