<s>[INST] General Linear Transformation 
Let $v_1 = \begin{bmatrix} 2 \\ -1  \end{bmatrix}$ and $v_2=\begin{bmatrix} 1 \\  -1 \end{bmatrix}$ and let $$A= \begin{bmatrix} 3 & 2 \\ -2 & 1 \end{bmatrix}$$ be a matrix for $T\colon \Bbb R^2\to \Bbb R^2$ relative to the basis $B = \{v_1, v_2\}$. 

From this I found that:

$T(v_1) = \begin{bmatrix} 4 \\ -1 \end{bmatrix}$ and $T(v_2) = \begin{bmatrix} 5 \\ -3 \end{bmatrix}$

How would I find a formula for $T\begin{pmatrix} \begin{bmatrix}x_1 \\ x_2\end{bmatrix} \end{pmatrix}$
The answer in my book is $\begin{bmatrix} -x_1-6x_2 \\ 2x_1+5x_2  \end{bmatrix}$
 [/INST] So, we should probably take the inverse of $\begin{pmatrix} 2&1 \\ -1&-1 \end{pmatrix}^{-1} = \begin{pmatrix} 1&1 \\ -1&-2 \end{pmatrix}$. This is called the change of basis. So, $\begin{pmatrix} 4&5 \\ -1&-3 \end{pmatrix}\cdot \begin{pmatrix} x_{1}\\-x_{1} \end{pmatrix} + \begin{pmatrix} 4&5 \\ -1&-3 \end{pmatrix}\cdot \begin{pmatrix} x_{2}\\-2x_{2} \end{pmatrix} = \begin{pmatrix} -x_{1}-6x_{2}\\2x_{1}+5x_{2} \end{pmatrix}$.
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