Linear transformation with respect to basis problem Let $T: \mathbb{R}^2 \to \mathbb{R}^2$ such that $T(\begin{bmatrix} 3 \\ 1 \end{bmatrix}) = \begin{bmatrix} 1 \\2 \end{bmatrix}$ and $T(\begin{bmatrix} -1 \\ 0 \end{bmatrix}) = \begin{bmatrix} 1 \\1 \end{bmatrix}$. Find the matrix $A$ representing $T$. 
I understand that to approach this problem, I have to view $\begin{bmatrix} 3 \\1 \end{bmatrix}$ and $\begin{bmatrix} -1 \\ 0 \end{bmatrix}$ as a basis, $B = \{ v_1, v_2 \}$, where $[v1 v2]$ is the transition matrix from $[x]B$ to $x$. How do I use $\begin{bmatrix} 1\\2 \end{bmatrix}$ and $\begin{bmatrix} 1 \\1 \end{bmatrix}$? I'm unclear on their connection to the basis vectors $v_1$ and $v_2$.

Relative to the bases $B$ and the standard basis,  the matrix is:. $\begin{pmatrix}1&1\\2&1\end{pmatrix}$.
The change of basis matrix is: $\begin{pmatrix}3&-1\\1&0\end{pmatrix}$.
The latter changes basis from $B$ to the standard basis.
Thus you want: $\begin{pmatrix}1&1\\2&1\end{pmatrix}\begin{pmatrix}3&-1\\1&0\end{pmatrix}^{-1}$.
I will leave this calculation for you.

 $\begin{pmatrix}1&1\\2&1\end{pmatrix}\begin{pmatrix}0&1\\-1&3\end{pmatrix}=\begin{pmatrix}-1&4\\-1&5\end{pmatrix}$

