How to find the linear transformation associated with a given matrix? Good day, I have a little doubt: It is well known that given two bases (or even one if we consider the canonical basis) of a vector space, every linear transformation $T:V \rightarrow W$ can be represented as a matrix, but since this is an isomorphism between $L(V,W)$ and $\mathbb{M}_{m\times n}$ where the latter represents the space $m\times n$ matrices on the same field in which are defined respectively vector spaces. That's where my question comes up, I know find the matrix associated with the linear transformation, but not know how to move from the matrix transformation, ie given any matrix, find the linear transformation that defines it. I wish you could please explain the theoretical process and to see a practical example. Thank you very much, I know it's definitely something silly, but I'm still a student.

Suppose that you have an $m,n$ matrix $A$. Choose a basis $B$ of $V$ and another one $B'$ of $W$. The linear transformation associated with $A$ relative to the bases $B$ and $B'$ is $T(v) = Av$, where $v$ is to be written as a column whose entries are the coefficients of $v$ in the basis $B$ and the resulting column $T(v)$ has entries which are the coefficients of $T(v)$ in the basis $B'$. If you choose other bases, you get different linear transformations 
