Gilbert Strang, Linear Algebra and it's applications Pg 129
Suppose the vectors $x_{1}, \ldots, x_{n}$ are a basis for the space $\mathbf{V}$, and vectors $y_{1}, \ldots, y_{m}$ are a basis for $\mathbf{W}$. Each linear transformation $T$ from $\mathbf{V}$ to $\mathbf{W}$ is represented by a matrix $A$. The $j$ th column is found by applying $T$ to the $j$ th basis vector $x_{j}$, and writing $T\left(x_{j}\right)$ as a combination of the $y$ 's: Column $j$ of $A \quad T\left(x_{j}\right)=A x_{j}=a_{1 j} y_{1}+a_{2 j} y_{2}+\cdots+a_{m j} y_{m} .$
I can understand its proof if the basis of $\mathbf V$ are the standard unit vectors, but can't see how this can be proven for a general basis set.
Can anyone please show me how to derive it? Thank you