16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 3.5.6

Linear Transformations

Mathematics — Linear Algebra · linear-transformation-via-functionals

Problem

Let mm and nn be positive integers and FF a field. Let f1,…,fmf_1, \ldots, f_m be linear functionals on FnF^n. For α\alpha in FnF^n define Tα=(f1(α),…,fm(α)).T\alpha = (f_1(\alpha), \ldots, f_m(\alpha)). Show that TT is a linear transformation from FnF^n into FmF^m. Then show that every linear transformation from FnF^n into FmF^m is of the above form, for some f1,…,fmf_1, \ldots, f_m.

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