Write a vector v in a basis given a set

Question

I am trying to write the vector $v = (2,3)$ in the basis outlined by the set $\{(−1, 1),(3, −5)\}$. I figured that I would take the inverse of the set matrix, and multiply it to v. However, this does not seem to be the solution. Any help would be greatly appreciated.

Answer 1

Define $v_1=(-1,1)$ , $v_2=(3,-5)$ and the vector of coefficients as $(a,b)$. Hence you must have $$v=(a,b)\cdot \begin{bmatrix}v_1\\v_2\end{bmatrix}$$which leads to $$v\cdot \begin{bmatrix}v_1\\v_2\end{bmatrix}^{-1}=(a,b)$$