For any complex vector bundle $E$ with dimension $2n$ over the even dimensions manifold $M$, do there exist 2n generic sections? (more explicitly, 2n sections $e_1,…,e_{2n}$ such that every $e_i$ intersect with the space spaned by {$e_1,…,e_{2n}$}-{$e_i$} transversely).
I try to do this by partition of unity and embedding $E$ to a trivial bundle, but I have no idea to control a section to make it transvers to the zero section.Could anyone help me?
Thank you for your answer!