Show that any linear subspace $U \subset V$ of dimension 2 corresponds to a unique line through the origin in $\Lambda^{2}(V)$

Question

Show that any linear subspace $U \subset V$ of dimension 2 corresponds to a unique line through the origin in $\Lambda^{2}(V)$

I am currently starting to learn about affine geometry and I am actually not so familiar with the wedge product. Any guidance would be much appreciated.

Answer 1

Say $\{e_1,e_2\}$ is a basis of $U$. The subspace with basis $e_1\wedge e_2$ is then a one-dimensional subspace of $\Lambda^2(V)$. This subspace is indeed only depending on $U$ since if you take any other basis of $U$, say the basis $\{u,v\}$, where $u=ae_1+be_2$ and $v=ce_1+de_2$, then $$ u\wedge v = (ae_1+be_2)\wedge (ce_1+de_2) = (ad-bc)e_1\wedge e_2, $$ where $(ad-bc)\ne 0$, since $\{u,v\}$ forms a basis of $U$.