Remark 1

Let V be a right vector space over the field D, of finite dimension $n$. We view the dual space $V^*=$ Hom$_D(V,D)$ as a right vector space over the opposite field $D^o$ of D. Let $u$ be an automorphism of V, and let $^tu$ be the automorphism of $V^*$ that is the transpose of $u$. If $u$ is represented by a matrix A in $\mathbf{M}_n(D)$ with respect to a basis $(e_1, . . . , e_n)$ of V, then the automorphism $^tu$ is represented by the matrix $^tA$ in $\mathbf{M}_n(D^o)$ with respect to the basis $(e^*_1, . . . , e^*_n)$ of $V^*$ dual to $(e_1, . . . , e_n)$ (II, §10, No. 4, p. 344, Proposition 3). By Example 7, the determinant of $^tu$ is equal to that of $u$.