Let $G$ be the Lie group of matrices of the form $\begin{pmatrix}a & b \\ 0 & 1\end{pmatrix}, a\ne 0, b\in\mathbb{R}$. Let $\phi(\begin{pmatrix}a & b \\ 0 & 1\end{pmatrix})=(a,b)$ be a coordinate chart. Describe the space of left invariant vector fields $L_X, X\in Lie(G)$ in term of $\phi$ coordinates.
The space of left invariant vector fields is simply the Lie algebra. I computed that $Lie(G)$ is the group of matrices of the form $\begin{pmatrix}a & b \\ 0 & 0\end{pmatrix}, a, b\in\mathbb{R}$ which is the Lie algebra generated by $A=\begin{pmatrix}1 & 0 \\ 0 & 0\end{pmatrix}$ and $B=\begin{pmatrix}0 & -1 \\ 0 & 0\end{pmatrix}$. How do I write this in terms of $\phi$-coordinates?
To find the left invariant vector fields we have to study the behavior of $l_A:G\to G, B\to AB$ because the left inv.v.f. satisfies the following equality: $$V\in\chi(G) \;\;s.t.\;\; d_el_AV_e=V_A\;\; \forall A\in G\;\;(\text{ with } e=Id). \;\;\;\; (*)$$ We know that, if $A=\left(\begin{matrix}a & b \\ 0 & 1\end{matrix}\right),B=\left(\begin{matrix}c & d \\ 0 & 1\end{matrix}\right)$: $$l_A B=\left(\begin{matrix}a & b \\ 0 & 1\end{matrix}\right)\left(\begin{matrix}c & d \\ 0 & 1\end{matrix}\right)=\left(\begin{matrix}ac & ad +b \\ 0 & 1\end{matrix}\right). $$
Now lets set the coordinate chart $\phi(\left(\begin{matrix}c & d \\ 0 & 1\end{matrix}\right))=(c,d)$, so the $d_el_A$'s matrix representation in this coordinates is: $$[d_el_A]_\phi=\left(\begin{matrix}\frac{\partial(ac)}{\partial c} & \frac{\partial(ac)}{\partial d} \\ \frac{\partial(ad +b)}{\partial c} & \frac{\partial(ad +b)}{\partial d} \end{matrix}\right)= \left(\begin{matrix}a & 0 \\ 0 & a\end{matrix}\right)\;\;\;(**)$$ Let be $V\in\chi(G)$ the general left invariant vector field (with $V_e=v_1\frac{\partial}{\partial c}_{|e}+v_2\frac{\partial}{\partial d}_{|e}$) and write it with respect to the vectors fields $\frac{\partial}{\partial c},\;\frac{\partial}{\partial d}$: $$V_A=f_1(A)\frac{\partial}{\partial c}_{|A}+v_2\frac{\partial}{\partial d}_{|A}\stackrel{(*)}{=} d_el_AV_e\stackrel{(**)}{=}av_1\frac{\partial}{\partial c}_{|A}+av_2\frac{\partial}{\partial d}_{|A}.$$