Given an $m\times n$ matrix $A$, there is an associated linear map $L_A: \mathbb{R}^n\longrightarrow\mathbb{R}^m$ as seen here. Conversely, given a linear map $L: \mathbb{R}^n\longrightarrow\mathbb{R}^m$ there exists an $m\times n$ matrix $A$ such that $L=L_A$. To see this, consider the unit column vectors $E^1,\cdots,E^n$ of $\mathbb{R}^n$. For each $j=1,\cdots,n$, let $L(E^j)=A^j$, where $A^j$ is a column vector in $\mathbb{R}^m$. For each $X\in\mathbb{R}^n$,
$$X=x_1E^1+\cdots+x_nE^n=\begin{pmatrix}
x_1\\
\vdots\\
x_n
\end{pmatrix}$$
and hence
\begin{align*}
LX&=x_1L(E^1)+\cdots+x_nL(E^n)\\
&=x_1A^1+\cdots+x_nA^n\\
&=AX
\end{align*}
where $A$ is the matrix whose column vectors are $A^1,\cdots,A^n$. The matrix $A$ is called the matrix associated with the linear map $L$.

Example. Let $L:\mathbb{R}^3\longrightarrow\mathbb{R}^2$ be the projection given by $$L\begin{pmatrix}
x\\
y\\
z
\end{pmatrix}=\begin{pmatrix}
x\\
y
\end{pmatrix}.$$
Find the matrix associated with $L$.

Solution. $$L(E^1)=\begin{pmatrix}
1\\
0\end{pmatrix},\ L(E^2)=\begin{pmatrix}
0\\
1
\end{pmatrix},\ L(E^3)=\begin{pmatrix}
0\\
0
\end{pmatrix}.$$
Thus, the matrix associated with $L$ is
$$A=\begin{pmatrix}
1 & 0 & 0\\
0 & 1 & 0
\end{pmatrix}.$$

Let us now consider a more general case. Let $V$ be a vector space and $\{v_1,\cdots,v_n\}$ a given basis of $V$. Let $L:V\longrightarrow V$ be a linear map. Then there exist numbers $c_{ij}$, $i,j=1,\cdots,n$ such that
\begin{align*}
Lv_1&=c_{11}v_1+\cdots+c_{1n}v_n,\\
&\vdots\\
Lv_n&=c_{n1}v_1+\cdots+c_{nn}v_n.
\end{align*}
Let $v=x_1v_1+\cdots+x_nv_n$. Then
\begin{align*}
Lv&=\sum_{i=1}^nx_iLv_i\\
&=\sum_{i=1}^nx_i\sum_{j=1}^nc_{ij}v_j\\
&=\sum_{j=1}^n(\sum_{i=1}^nx_ic_{ij})v_j.
\end{align*}
Hence, we have the following theorem.

Theorem. If $C=(c_{ij})$ is the matrix such that $Lv_i=\sum_{j=1}^nc_{ij}v_j$ and $X=\begin{pmatrix}
x_1\\
\vdots\\
x_n
\end{pmatrix}$ is the coordinate vector of $v$, then the coordinate vector of $Lv$ is ${}^tCX$ i.e. the matrix associated with $L$ is ${}^tC$ with respect to the basis $\{v_1,\cdots,v_n\}$.

Example. Let $L: V\longrightarrow V$ be a linear map. Let $\{v_1,v_2,v_3\}$ be a basis of $V$ such that
\begin{align*}
L(v_1)&=2v_1-v_2,\\
L(v_2)&=v_1+v_2-4v_3,\\
L(v_3)&=5v_1+4v_2+2v_3.
\end{align*}
The matrix associated with $L$ is
$$\begin{pmatrix}
2 & 1 & 5\\
-1 & 1 & 4\\
0 & -4 & 2
\end{pmatrix}.$$