Let $B$ be the subgroup of upper triangular matrices in $GL_2(\mathbb{R})$.
What are the left cosets of $B$ in $GL_2(\mathbb{R})$?
I was told that the set of all such cosets can be naturally put in one-to-one correspondence with the real projective line.
So, here's what I'm thinking... Suppose that my cosets are $aB$ for all $a\in A\subseteq GL_2(\mathbb{R})$. The real projective line is the set of all nonzero vectors in $\mathbb{R}^2$ with the equivalence relation $(x,y)\sim(x',y') \iff (x,y)=\lambda (x',y')$ for some $\lambda\in\mathbb{R}^*$. Then the set $A$ should consist of matrices allowed to range freely over the reals in two of their entries, but these two entries cannot both be zero. Furthermore, scaling these two entries results in the same coset.
One of the most natural situations in which one encounters cosets of a subgroup $H$ of $G$ is that when $G$ has some action on a set $X$ for which $H$ is the stabiliser subgroup of some specific element $x_0\in X$, in other words $H=\{\, g\in G\mid g\cdot x_0=x_0\,\}$. Then the cosets $gH$ of $H$ correspond precisely to the elements of the orbit of$~x_0$ in the action, under the correspondence $gH\mapsto g\cdot x_0$. The main point is that this is well defined: another element $gh$ of the same coset as$~g$ will have $gh\cdot x_0=g\cdot x_0$, so no conflicting definition results.
Now your subgroup $B$ is naturally interpreted as the stabiliser of the point $x_0=\langle\binom10\rangle$ of the projective line in the natural action of $GL_2(\Bbb R)$ on the projective line. Moreover this action is transitive, so the orbit of $x_0$ is the whole projective line. One easily checks the above general correspondence gives what you are after in this case.