# Orientation for a vector spaces determines a canonical orientation for the dual vector space

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.

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 $$det(A^t)=det(A)$$.