Numbers, vectors, matrices
Recall that the set of real numbers is $\mathbb{R}$ and that a vector , $v \in \mathbb{R}^n$ is just an n-tuple of numbers.\\
Similarly, an $n x m$ matrix is just a table of numbers, with n rows and m columns and we can write
$$A_{mn} \in \mathbb{R}^{mn}$$
Note that a vector is normally considered equivalent to a $n\times1$ matrix i.e. we view these as column vectors.
Examples
\begin{xmpl}
In R, a vector can be generated with:
\begin{lstlisting}
X<- 3:6
X
[1] 3 4 5 6
\end{lstlisting}
A matrix can be generated in R as follows,
\begin{lstlisting}
matrix(X)
[,1]
[1,] 3
[2,] 4
[3,] 5
[4,] 6
\end{lstlisting}
\begin{notes}
We note that R distinguishes between vector and matrices.
\end{notes}
\end{xmpl}