The background
I encounter this problem when I try to analyze the planar transformation of a 2D triangle. We ignore the translational shift in this problem.
Consider a 2D triangle whose edge vectors are $\vec x_0, \vec x_1\in \mathbb R^2$, these two vectors are coming from the same vertex, equivalent to the origin.
Apply an arbitrary transformation $T \in \mathbb R^{2 \times 2}$ on this triangle and we get $\vec x_0', \vec x_1'$, which construct a new triangle. $$ [\vec x_0', \vec x_1'] = T [\vec x_0, \vec x_1] $$
My thinking
Then I just think: a transformation from $x_0$ to $x_0'$ is simply $x_0' = s_0 R_0 x_0$, where $s_0$ is a scaling factor and $R_0$ a rotation matrix. so that I can get: $$ \begin{cases} \vec x_0' = s_0 R_0 x_0 \\ \vec x_1' = s_1 R_1 x_0 \\ \end{cases} $$
My question
The transformation $T$ has 4 DOFs. and $s_0, s_1, R_0, R_1$ also has 4 DOFs. So it is possible to build a relationship between them.
What is the relationship between this two stuff? How to parameterize $T$ in $s_0, s_1, R_0, R_1$? which is $$ T = T(s_0, s_1, R_0, R_1) ? $$