Let $V$ be a vector space with basis $e_1, \ldots, e_n$ and $V^*$ be its dual space with dual basis $e_1^*, \ldots, e_n^*$. Let $k$ be an integer between $1$ and $n$. Why $\wedge^{n-k}V=\wedge^{k}V^*$? Thank you very much.
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This is slightly false. The two are isomorphic, but not canonically so. There is a natural pairing $\Lambda^{n-k} V \times \Lambda^k V \to \Lambda^n V$ given by exterior product, but this pairing does not identify $\Lambda^{n-k} V$ with $(\Lambda^k V)^{\ast}$ until you pick an isomorphism $\Lambda^n V \simeq k$; this implies a choice of orientation, but is slightly stronger; one might say it implies a choice of "volume form." But it does not imply a choice of inner product. The (canonical) isomorphism between $(\Lambda^k V)^{\ast}$ and $\Lambda^k V^{\ast}$ comes from the way duals commute with tensor products. It should look pretty straightforward with a specific basis. |
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