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Suppose $V$ is an $n$-dimensional real vector space, with $n > 0$. Show that an orientation for $V$ determines a canonical orientation for $V^*$, the dual of $V$.

The idea I had in mind to show this is to take two ordered bases $\{v_i\}$ and $\{w_i\}$ for $V$ which induce the same orientation (i.e., there exists a matrix $M$ such that $Mv_i = w_i$ and $\det M > 0$) and look at the bases $\{ f^{\alpha}\}$ and $\{ g^{\alpha} \}$ on $V^*$ canonically determined by our bases on $V$ via the relations $f^{\alpha}(v_i) = \delta^{\alpha}_i$ and $g^{\alpha}(w_i) = \delta^{\alpha}_i$. However, I don't really know how to determine the change of basis matrix between $f^{\alpha}$ and $g^{\alpha}$. Any help would be great.

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Hint: if $A(v_i)=w_i, A^t(w_i^*)=v_i^*$ where $A^t$ is the transpose matrix where $(v_i^*)$ is the dual basis of $(v_i)$ and $det(A^t)=det(A)$.

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  • $\begingroup$ Ah of course. I’ve forgotten my basic linear algebra. Thanks! $\endgroup$ – SubSpace626 Aug 21 '19 at 9:49

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