Linear Transformations - Linear Algebra I have a problem with this exercise.
I think it can be solved with the properties of linear transformations [ T(v+w) = T(v) + T(w) and T(cv) = cT(v) ] but I'm not sure if it's valid and how to use it in this context.   

Does anyone have any idea how to solve it?  

Exercise

//Here is the exercise if you can´t see the image//     

Let V be a vector space and let T : V → V a linear transformation such that T($2v_1-3v_2$) = $5v_1 + 3v_2$ and T($−3v_1+5v_2$) = $4v_1 + 3v_2$. Write T($v_1$)  and  T($v_2$) in terms of $v_1$ and $v_2$.
The matrix $\begin{pmatrix}2 &amp; -3\\-3 &amp; 5\end{pmatrix}$ transforms coordinates with respect to basis $v_1, v_2$ 

to those with respect to basis $2v_1-3v_2, -3v_1+5v_2$.  

The matrix $\begin{pmatrix}5&amp;4\\3&amp;3\end{pmatrix}$ represents $T$ with coordinates with respect to $2v_1-3v_2, -3v_1+5v_2$ in the domain 

and $v_1,v_2$ for the range.

Therefore, the columns of $\begin{pmatrix}5&amp;4\\3&amp;3\end{pmatrix}\begin{pmatrix}2 &amp; -3\\-3 &amp; 5\end{pmatrix}^{-1}$ give the images of $v_1$ and $v_2$ under $T$.
Hint:

$$v_1=5(2v_1-3v_2)+3(-3v_1+5v_2)$$

$$v_2=3(2v_1-3v_2)+2(-3v_1+5v_2)$$