I have read some of the other questions here on change of basis, but I am still not sure if I am solving my problem correctly. I would really appreciate it if someone could help me. Thanks.
Consider $S_1, S_2 \subset \mathbb{R}^2$. The local coordinate bases for $S_1$ are $\{\mathbf{e}_2,-\mathbf{e}_1\}$, and for $S_2$ are $\{\mathbf{e}_1, \mathbf{e}_2\}$, with $\mathbf{e}_i$ the standard bases. Now, consider maps, $$ \phi_i : S_i \rightarrow \mathbb{R}^2 $$ $$ (u_1, v_1) \mapsto u_1\mathbf{\alpha}_2 + v_1\mathbf{\alpha}_3 $$ $$ (u_2, v_2) \mapsto u_2\mathbf{\alpha}_1 + v_2\mathbf{\alpha}_2 $$ where $\alpha_i$ are vectors in $\mathbb{R}^2$. The maps are such that, $$ \phi_1([0, 1]\times\{0\}) = \phi_2(\{0\}\times[0, 1]). $$
Next, given the coordinates $[u_1, v_1]^T \in S_1$, I want the transformation map from $S_1$ to $S_2$. My first thought was, $$ [\alpha_2, \alpha_3]\left[ \begin{array}{c} u_1\\ v_1 \end{array}\right] = \left[ \begin{array}{c} a\\ b \end{array}\right] = [\alpha_1, \alpha_2]\left[ \begin{array}{c} u_2\\ v_2 \end{array}\right] $$ and so the matrix representing the transformation from $S_1$ to $S_2$ would be, $$ [\alpha_1, \alpha_2]^{-1}[\alpha_2, \alpha_3]. $$
However, my teacher's notes indicate that the correct transformation is, $$ S[\alpha_2, \alpha_3]^{-1}[\alpha_1, \alpha_2]S, $$ with, $$ S = \left[ \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right]. $$ What am I doing wrong? Thank you for your time.