I want to study the Lie group of affine transformations of the form \begin{bmatrix}a&b&x\\c&d&y\\0&0&1\end{bmatrix}. When I parameterize it to be of the form \begin{bmatrix}a*cos(t)&-b*sin(t)&x\\a*sin(t)&b*cos(t)&y\\0&0&1\end{bmatrix}, it defines an intrinsically 5-dimensional Lie subgroup, am I right with that? How can I determine the tangential space of the subgroup? And how do I get the adjoint representation on the Lie group and the Lie algebra? Thank you very much and best regards
Antoni