Basis changing for transformations matrices in linear algebra

Question

I can't understand the process of how I can find transformation matrices in any bases other than the standard basis.

$T(x,y)=(x,2x+y)$ and the basis B is $\vec{v_{1}}=(1,1)$, $\vec{v_{2}}=(-2,1)$

I am familiar with finding a vector for another basis but I can't get understand the process for matrices

Answer 1

Since$$T(v_1)=(1,3)=\frac73v_1+\frac23v_2$$and$$T(v_2)=(-2,-3)=-\frac83v_1-\frac13v_2,$$the matrix that you're after is$$\begin{bmatrix}\frac73&-\frac83\\ \frac23&-\frac13\end{bmatrix}.$$