action of general linear group on the set of lines through origin

Question

Let $A $ denotes the set of all lines through the origin.Let $G=GL_{n}(\mathbb{R})$ be the general linear group of $ 2 \times 2 $ matrices and $H$ be the subgroup of $G$ containing all lower triangular matrices. Then find a line $l$ in $A$ whose stabilizer in $G$ is exactly H , i.e; $ stab_{G}(a)=H$. $$$$ I have tried in this way, If a group $G$ acts on a set $X$ then for $ x \in X$ , $stab_{G}(x)=\{g \in G : gx=x \}$. Replacing $ X$ by $A \ and \ x \ by \ l$ , we get $ stab_{G}(l)=\{g \in G: gl=l \}$. In order to show that $ Stab_{G}(l)=H$, we have to show that for any $ s \in Stab_{G}(l) \implies s \in H$. Right here i can't proceed further. please help me from here, thanks.