I forgot how to compute the transformation in a given basis. :'(
For example, say I have the transformation \begin{equation}(a, b) \mapsto \begin{bmatrix}10a - 6b \\ 17b - 10b \end{bmatrix}.\end{equation}
The basis in $\mathbb{R}^2$ is made up of the vectors $(1, 2)$ and $(2, 3)$.
What's the procedure again?
For one, I am assuming you meant $17a - 10b$ for the second entry.
The transformation in terms of a given basis is just the image of that basis when written in standard coordinates.
Then, if I read this correctly, $(1,0)^{T} \rightarrow (10, 17)^{T}$ and $(0,1)^{T} \rightarrow (-6, -10)^{T}$ so that the image of the basis $\{\left( \begin{array}{c} 1 \\ 2 \end{array} \right),\left( \begin{array}{c} 2 \\ 3 \end{array} \right)\}$ is $\{\left( \begin{array}{c} -2 \\ -3 \end{array} \right),\left( \begin{array}{c} 2 \\ 4 \end{array} \right)\}$ making $\left( \begin{array}{cc} -2 & 2 \\ -3 &4 \end{array} \right)$ the matrix representing the transformation in terms of the basis given.