every linear transformation is associated with a matrix and vice versa I'm quite confused with this theorem
Can someone explain me step by step how do I find a linear transformation from a given matrix and finding a matrix from a given linear transformation please?
Examples will be great.
Thanks alot.
 A: Ok given a matrix Assuming it to be  $ n \times n = (a_{ij})$
Now let V be a vector space of dimension n over a field  where the matrix has its values. Now let $ v_1,v_2,..v_n$ be a basis of V.
Define $ A : V \to V$ by $A(v_i) = \sum_{j=1}^n a_{ji} v_j$.
Conversely given a map from V to V. define a matrix whose entries are $ a_{ij}$ where $a_{ij}$ is the coefficient of $v_j$ in the expansion of  $A(v_i)$in terms of $v_i$'s.
A: Let 
$$A=\begin{pmatrix} 1 & 0 & 9 \\ 0 & -2 & 1 \\ 0 & 0 & 3 \end{pmatrix}\in\Bbb R^{3\times 3}$$
be a matrix. The linear map associated to it is the map
\begin{align*}
F_A:\Bbb R^3 &\longrightarrow \Bbb R^3 \\ 
 \begin{pmatrix} a\\b\\c\end{pmatrix} &\longmapsto 
\begin{pmatrix} 1 & 0 & 9 \\ 0 & -2 & 1 \\ 0 & 0 & 3 \end{pmatrix}
\cdot\begin{pmatrix} a\\b\\c\end{pmatrix}
 = 
 \begin{pmatrix} a+9c\\-2b+c\\3c\end{pmatrix}.
\end{align*}
Conversely, assume you have any linear map $F:\Bbb R^3\to \Bbb R^3$. Let
\begin{align*}
e_1 &=  \begin{pmatrix} 1\\0\\0\end{pmatrix}, &
e_2 &=  \begin{pmatrix} 0\\1\\0\end{pmatrix}, &
e_3 &=  \begin{pmatrix} 0\\0\\1\end{pmatrix}.
\end{align*}
Let 
$$
\begin{pmatrix} A_{1i}\\A_{2i}\\A_{3i}\end{pmatrix} := F(e_i),
$$
this defines a Matrix $A\in\Bbb R^{3\times 3}$ with entries $A_{ij}$. 
If you apply this to our $F_A$ above, you see how we get the matrix $A$ back. All of this generalizes as explained in the other answer.
