Tautological vector bundle over $G_1(\mathbb{R^2})$ isomorphic to the Möbius bundle > Let $V$ be a finite dimensional vector space, and let $G_k(V)$ be the Grassmannian of $k$-dimensional subspaces of $V$. Let $T$ be the disjoint union of all these $k$-dimensional subspaces and let $\pi:T\rightarrow G_k(V)$ be the natural map sending each point $x \in S$ to $S$. Then $T$ has a unique smooth manifold structure making it into a smooth rank-$k$ vector bundle over $G_k(V)$, with $\pi$ as a projection and with the vector space structure on each fiber inherited from $V$. $T$ is called the **tautological vector bundle over** $G_k(V).$ What I want to prove is that tautological vector bundle over $G_1(\mathbb{R^2})$ is isomorphic to the Möbius bundle. (This is a problem from _Introduction to Smooth Manifolds_ by Lee and Möbius bundle is defined as in Lee's book, page 105. Also I took the definition of the tautological vector bundle over $G_k(V)$ from Lee's book as well.)

As @Sam pointed out $G_1(\mathbb{R^2}) \cong \mathbb{RP^1} \cong \mathbb{S^1}$. Now, writing a smooth bundle isomorphism is not so hard.