Let $G = \left\{ \mathrm{exp}\, t \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \right\}$. I want to show that $G$ is a Lie subroup of $\mathrm{GL}(\Bbbk)$, where $\Bbbk = \mathbb{R}$ or $\mathbb{C}$. To do so I have to show that the $G$ can be presented, in any neighberhood of its point, by a system of equations such that this system consists of differentible functions and its Jacobian has a rank equal the number of the equations.
My truble is to find such system. It is clear that $M(t)^n = \begin{pmatrix} t^n & nt^n\\0 &t^n \end{pmatrix}$ for any $n\ge 0$, where $M(t) = \begin{pmatrix} t & t \\ 0 &t \end{pmatrix}$, and thus $\mathrm{exp}(M(t)) = \begin{pmatrix} e^t & te^t \\ 0 & e^t \end{pmatrix}$ because of $\sum_{k=1}^\infty t^k/(k-1)! = t\sum_{k=1}^\infty t^{k-1}/(k-1)! = te^t$. And I don't know what is the next step. Of course I can say that $G = \begin{pmatrix} g_{11} & g_{12} \\ g_{21} & g_{22} \end{pmatrix}$ with $\begin{cases} g_{11} - t g_{12} = 0 \\ g_{21}=0 \\ g_{11} - g_{22} =0 \\ g_{11} - e^t = 0 \end{cases}$ and the Jacobian is $\begin{pmatrix} 1 & -t & 0 & 0 \\ 0 & 0 & 1 & 0\\ 1 &0 &0 & 0 -1\\ 1 & 0 & 0 & 0 \end{pmatrix}$. But I am sure that it is the right answer.
