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I am reading the book Calculus on Manifolds by Spivak and I am trying to solve problem 4.1(b):

Let $e_1, \dots, e_n$ be the usual basis on $\mathbb{R}^n$ and let $\varphi_1, \dots, \varphi_n$ be the dual basis. Show that $\varphi_{i_1}, \wedge \cdots \wedge \varphi_{i_k}(v_1, \dots, v_k)$ is the determinant of the $k \times k$ minor of $(v_1 \, \cdots \, v_k)^T$ obtained by selecting columns $i_1, \dots, i_k$.

Let $$v_r = \sum_{s = 1}^n \alpha_{rs} e_s.$$ Then \begin{align*} \varphi_{i_1} \wedge \cdots \wedge \varphi_{i_k}(v_1, \dots, v_k) &= \sum_{\sigma \in S^k} \mbox{sgn } \sigma [\varphi_{i_1}(v_{\sigma(1)}) \cdots \varphi_{i_k}(v_{\sigma(k)})]\\ &= \sum_{\sigma \in S^k} \mbox{sgn } \sigma [ \alpha_{\sigma(1), i_1} \cdots \alpha_{\sigma(k), i_k}]. \end{align*}

How should I relate the determinant to this expression? Thank you

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The space of alternating $k$-forms on a $k$-dimensional vector space is a vector space of dimension $1$. Show that this is such a form and calculate it's value on the standard basis.

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  • $\begingroup$ Actually I am trying to use this to show the linear independence of the basis $\{\varphi_{i_1} \wedge \cdots \wedge \varphi_{i_k} : 1 \leq i_1 < \cdots < i_k \leq n\}$. Are there any other approaches? $\endgroup$ – s20012303 Apr 21 '17 at 13:36
  • $\begingroup$ @s20012303 You could check whether the result of your computation relates to Leibnitz' formula for the determinant: en.wikipedia.org/wiki/Leibniz_formula_for_determinants $\endgroup$ – Thomas Apr 21 '17 at 13:44

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