Kunze Linear Algebra · Problem 5.3.11

Determinants

Mathematics — Methods · determinant-linear-operator

Problem

Let TT be a linear operator on FnF^n. Define DT(α1,…,αn)=det⁡(Tα1,…,Tαn).D_T(\alpha_1, \ldots, \alpha_n) = \det(T\alpha_1, \ldots, T\alpha_n). ((a)) Show that DTD_T is an alternating nn-linear function. ((b)) If c=det⁡(Tϵ1,…,Tϵn)c = \det(T\epsilon_1, \ldots, T\epsilon_n), show that for any nn vectors α1,…,αn\alpha_1, \ldots, \alpha_n we have det⁡(Tα1,…,Tαn)=c det⁡(α1,…,αn).\det(T\alpha_1, \ldots, T\alpha_n) = c\,\det(\alpha_1, \ldots, \alpha_n). ((c)) If B\mathfrak{B} is any ordered basis for FnF^n and AA is the matrix of TT in the ordered basis B\mathfrak{B}, show that det⁡A=c\det A = c. ((d)) What do you think is a reasonable name for the scalar cc?

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