I know how to find the orthogonal and the special linear group of $2$ by $2$ matrices. This is because I know their “defining” properties. How can I find the Lie algebra of:
$$A = \left(\begin{array}[c c] - a_1 & a_2\\ 0& a_1^2 \end{array}\right),$$
Where the matrices are invertible? I don’t know their defining characteristic that is my big problem. If I know it, I can use dedicated to solve. Maybe it has to do something with their matrix potential or their trace?
Please help.
Let $f:Gl(2,\mathbb{R})\rightarrow \mathbb{R}$ defined by $f(\pmatrix{a&b\cr c&d})=c$, $f$ is a submersion, we deduce that $f^{-1}(0)$ the set of invertible upper triangular matrices $U(2,\mathbb{R})$ is a closed submanifold and a Lie subgroup of $Gl(2,\mathbb{R})$. This implies that the Lie algebra of $U(2,\mathbb{R})$ is the kernel of the differential of $f$ at the identity which is the subset of upper triangular matrices which are not necessarily invertible.