I'm trying to do exercise 2.2.2 in Huybrecths's Complex Geometry: An Introduction. It reads:
2.2.2 Show that any short exact sequence of holomorphic vector bundles $0\to L\to E\to F\to 0$, where $L$ is a line bundle, induces a short exact sequence of the form $$0\to L\otimes \bigwedge\nolimits^{i-1} F \to \bigwedge\nolimits^{i}E\to \bigwedge\nolimits^{i} F\to 0.$$
I think my main struggle is how to work with exterior powers (and the other constructions) of vector bundles. I know that, for example, $\bigwedge\nolimits^i E$ has (canonically, whatever that means) fiber $\bigwedge\nolimits^i E(x)$ over $x$ but I'm having a hard time then thinking what $E$ itself is (i.e. how these fiber come together in a smooth way).
I figure that maybe it suffices to work with fibers though, i.e. to show that the induced $$0\to L(x) \otimes \bigwedge\nolimits^{i-1} F(x) \to \bigwedge\nolimits^{i}E(x)\to \bigwedge\nolimits^{i} F(x)\to 0$$ is exact for all $x$. Would this be enough and if so, why is it enough?
Anyway, if it is enough, here is my attempt. Denote the initial maps by $\phi:L\to E$ and $\psi:E\to F$. The map $\bigwedge\nolimits^{i} E(x) \to \bigwedge\nolimits^{i} F(x)$ is defined on basis elements by $s_1\wedge \cdots \wedge s_i \mapsto \psi_x(s_1)\wedge \cdots \wedge \psi_x(s_i).$
Now we want a map $L(x)\otimes \to \bigwedge\nolimits^{i-1}F(x)\to \bigwedge\nolimits^{i} E(x)$. Now $L(x)$ is $1$-dimensional as $L$ is a line-bundle so say $L(x)$ is spanned by $v$. Then we can map $v$ to $E(x)$ via $\phi_x$. It remains to map an element of the form $t_1\wedge \cdots \wedge t_{i-1}$ for $t_i\in F(x)$. Now as the map $E(x)\mapsto F(x)$ is surjective, there exists $e_j\in E(x)$ with $\psi_x(e_j)=t_j$. Now I would like to map $t_1\wedge \cdots \wedge t_{i-1}$ to $e_1\wedge \cdots \wedge e_{i-1}$ but this seems to involve many choices and I'm not sure it's correct. I do know that if $\psi_x(e_j')=t_j$ too then $e_j, e_j'$ differ by a multiple of $\varphi_x(v)$ because of the exactness of the initial sequence but I'm not sure how to use this.
So could someone please clarify some things here, in particular: (1) why is it enough to work on the fibers (and if it's not enough, how would one approach this problem) and (2) how to define that last map properly.
Thanks in advance!