What is the difference between linear operator in a pair of basis and a linear operator? Let $e_1,...,e_n$ be a basis in $X$ and $f_1,...,f_m$ basis in $Y$. From the following image the matrix $A$ is a linear operator in a pair of basis.

What does this mean?
 A: Start with a linear operator $A : X \to Y$.  To get a matrix that represents the operator, we need to use bases for the spaces.
If $e_1,\dots, e_n$ is a basis for $X$ and $f_1, \dots, f_m$ is a basis for $Y$, then we do this:
For each $j = 1, \dots n$, the image $A(e_j)$ is a vector in $Y$, so it has an expansion in terms of $f_1,\dots, f_m$.  Let that expansion be
$$
A(e_j) = a_{1j}f_1+\dots + a_{mj}f_m
$$
The matrix corresponding to $A$ is then the matrix of all these coefficients.  Column $j$ of the matrix corresponds to $A(e_j)$, the entry in row $i$ is $a_{ij}$; that is the coefficient of $f_i$ in the expansion of $A(e_j)$.
A: Given a linear map $A$, the lines
$$
A(e_1) = a_{11}f_1 + a_{21}f_2 + \cdots + a_{m1}f_m\\
A(e_2) = a_{12}f_1 + a_{22}f_2 + \cdots + a_{m2}f_m\\
\vdots\\
A(e_n) = a_{1n}f_1 + a_{2n}f_2 + \cdots + a_{mn}f_m
$$
completely define what the linear map $A$ does to any vector in $X$, as the $e_n$ are a basis for $X$. Which is to say, if you know it's written in this form, and you know that we're using the $e_i$ basis for $X$ and the $f_j$ basis for $Y$, then the numbers $a_{ij}$ that appear in those lines completely determine what the linear map $A$ does to any vector in $X$.
So we put those numbers $a_{ij}$ in a neat little table, and we call such a table a "matrix". And since this particular linear map is called $A$, and the matrix is constructed directly from the linear map, we often give the matrix the name $A$ as well.
And the entire act of taking a vector $v\in X$, write it out as a linear combination of the $e_i$, and then using the $a_{ij}$ as described in the formula above to find $A(v)$ expressed as a linear combination of the $f_j$'s, we call matrix-vector multiplication, and we write it $Av$.
Certainly, we can go in the reverse as well: Given any $m\times n$ matrix of numbers, we can by this rule interpret that matrix as a linear map.
So what this paragraph does is, it explains the connections between linear maps and matrices: Given two vector spaces, $X$ of dimension $n$ and $Y$ of dimension $m$, each with a chosen basis, there is always a nice correspondence between the set of $m\times n$ matrices and the set of linear maps $X\to Y$.
