Find matrix representation based on two vector transformations Find the matrix representation of the linear transformation T that maps the input vector $\vec{x}_1 = (1,1)^T$ to the output vector $\vec{y}_1 = (2,-3)^T$ and maps the input vector $\vec{x}_2 = (1,2)^T$ to the output vector $\vec{y}_2 = (5,1)^T$
I am given a hint that this is a "non-standard basis probing" question.
Independently I see the transformations seem to be $T_1 = (1,-4)$ and $T_2 = (4,-1)$
However I can't seem to create the correct matrix based on those two independent transformations. Any hints?
 A: Hint: suppose the matrix is $V$, then we have 
$$
\begin{bmatrix}
x_1&
x_2\\
\end{bmatrix} V = 
\begin{bmatrix}
1 & 1\\
1 & 2\\
\end{bmatrix} V = \begin{bmatrix}
2 & 5\\
-3 & 1\\
\end{bmatrix} =  \begin{bmatrix}
y_1& y_2\\
\end{bmatrix}
$$
Consider to find the inverse matrix of $\begin{bmatrix}
x_1&
x_2\\
\end{bmatrix}$
A: You can write this as a matrix equation. Create a matrix $X$ from $\vec x_1$ and $\vec x _2$,and another one $Y$ from $\vec y_1$ and $\vec y_2$. Then $TX=Y$. Multiply both sides on the right with $X^{-1}$ and you get the answer.
$$T=\begin{pmatrix}
2 & 5 \\
-3 & 1
\end{pmatrix}\begin{pmatrix}
1 & 1 \\
1 & 2
\end{pmatrix}^{-1}$$
A: You know the matrix representation of $T$ under the basis $(1, 1)^T$ and $(1,2)^T$. 
Use a change of basis matrix. If $A$ sends $(1,1)^T$ to $(1,0)^T$, and $M$ sends $(1,0)^T$ to $(2,-3)^T$, then their composition sends $(1,1)^T$ to $(2,-3)^T$, and similar for the other basis vector
M is easy to calculate - you know what it sends the standard basis to. $A$ is a bit tricky - it sends $(1,1)^T$ to $(1,0)^T$ and $(1,2)^T$ to $(0,1)^T$. 
You know what $A^{-1}$ sends the standard basis to. You can then construct the matrix for $A^{-1}$, invert it to get the matrix for $A$, and then multiply it with $M$. (in the order corresponding to doing A first, then M)
