Most textbooks represent vectors (columns) of a tabular dataset $X$ as:
$$X_j=\begin{bmatrix}
    x_{1j} \\
    x_{2j} \\
    x_{3j} \\
    \vdots \\
    x_{nj} \\
\end{bmatrix}$$
So, the dataset can be represented as $$X=\begin{bmatrix}
    X_{1} & X_{2} & X_{3} & \dots  & X_{p}
\end{bmatrix}$$.

University of Michigan - Ann Arbor

A datapoint, instance, sample, or observation, is a unit of data in a dataset.
When datasets are tabular, $x_{ij}$ usually represents the value of the $j$th variable (attribute) in the $i$th data point, where $i=1,2,...,n$ indicates the index of the variable and $j=1,2,...,p$ identifies the variable. This way, a dataset can be represented as an $n\times p$ matrix X, where the row (or vector) $x_i$ represents a datapoint that consists of the $p$ values of $x_{i1}, x_{i2}, x_{i3}, \dots , x_{ip}$ corresponding to each variable.
$$X=\begin{bmatrix}
    x_{11} & x_{12} & x_{13} & \dots  & x_{1p} \\
    x_{21} & x_{22} & x_{23} & \dots  & x_{2p} \\
    \vdots & \vdots & \vdots & \ddots & \vdots \\
    x_{n1} & x_{n2} & x_{n3} & \dots  & x_{np}
\end{bmatrix}
=\begin{bmatrix}
    x_1^T \\
    x_2^T \\
    \vdots \\
    x_n^T \\
\end{bmatrix}$$ 

Datapoint

The purpose of a statistical learning model is to use a statistical model and the training data to find the function $\hat{f}$ as an estimate of the unknown function $f$ that maps the matrix of input variables $X$ to the outcome vector $Y$, as best as possible.
$$Y = f(X) + \epsilon \approx \hat{f}(X)$$ 

Purpose of supervised statistical learning

James, G., Witten, D., Hastie, T., & Tibshirani, R. (2013). An introduction to statistical learning (Vol. 112, pp. 3-7). New York: springer.

An Introduction to Statistical Learning with Applications in R

Formulating the dataset in supervised statistical learning

Data points most fall into one of two categories:
- Qualitative
- Quantitative

Types of Data Points

A supervised statistical learning model assumes:
1. A relationship between the matrix of input variables $X$ and the outcome $Y$.
2. Randomness and independence of $\epsilon$ from $X$.
3. $\epsilon$ should have a mean of 0.

Supervised statistical learning assumptions

The error term $\epsilon$ indicates the residual of fitted values compared to the observed outcomes, i.e., $\epsilon$ accounts for variation in $Y$ that cannot be explained by the estimated function $f$:
$$Y=f(X)+\epsilon$$ 

$\epsilon$ in supervised statistical learning

After choosing a functional form for $\hat{f}$, based on our domain knowledge or other techniques, we need to estimate the parameters of the model based on the training data. The process of finding optimal values for the parameters of $\hat{f}$ based on the training data is called "training" or "fitting" the model.

Training or Fitting a supervised statistical learning model

In supervised learning problems, a label, also known as a target, is a special attribute of an example that a model aims to predict. It represents the known outcome or ground truth that is not part of the model's input features.

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