I'm trying to do exercise 2.2.3 on Huybrechts, Complex Geometry:
Show that for any holomorphic vector bundle $E$ of rank $r$ there exists a non-degenerate pairing $$\Lambda^k E\times \Lambda^{r-k} E\to \Lambda^r E.$$
I don't understand what is the meaning of pairing in this context, because $\Lambda^k E\times \Lambda^{r-k} E$ is not a bundle.
For vector spaces, it is relatively easy to get a non-degenerate pairing $\Lambda^k V\times \Lambda^{r-k} V\to \Lambda^r V$ via $(v_1\wedge\ldots\wedge v_k, w_1\ldots\wedge w_{r-k})\mapsto (v_1\wedge\ldots\wedge w_{r-k})$ but I don't know how to translate this to the vector bundles.
Any comment will be appreciated. Thanks!